Logan already gave a great answer to the biological differences of the reproduction itself. I want to address the evolutionary psychology and how it would in term affect cultural differences, as I believe there will be some significant ones.
Cheaters sometimes Prosper
For the record whenever I say male or female please interpret that as meaning 'the herm who played the role of male (providing sperm) or female (birthed the child) in a given mating', it makes the answer much cleaner to not have to constantly add that qualifier
One issue that has shaped human culture and laws since their onset unfortunately still plagues our poor herms, and that is the possibility of infidelity. The male can never know with 100% confidence that he was not cheated on by his partner. Thus one would generally trust the child that came out of their womb as being theirs more then they trust a child birthed by another herm who could have theoretically mated with a different male and have birthed that male's child instead of yours.
This effect would start as one of evolutionary psychology. A hem will instinctually prefer investing resource on the child that definitely shares their genetics over investing resources on a child that potentially may not share their genetics.
How significant this psychological inclination to prefer children you birthed is depends on how assured the male is that he is the father of a child. If the animals were polyandrous or polygynandrous (ie the herm playing the role of female regularly mates with multiple males) then a herm would dedicate most of their resources to the children they birthed since there is a very real possibility they didn't father the child of a herm they mated with. If a species instead split off in such a way that a mongomous couple was isolated from all other herms during mating time so that it would be difficult for a partner to find someone to cheat with then the male is far more likely to treat children equally since he is relatively confident of the paternity of the child.
Just because a species is 'monogamous' does not mean this doesn't matter, because every monogamous species every studied still had regular cases of extra-pair copulation (ie cheating), simply put cheating is too effective an evolutionary strategy for it to never occur in a species. For humans the frequency of non-paternity events (ie the guy who thinks he is the father isn't) seems to be somewhere around 2-4% in modern history. I would presume that number would be higher in prehistory, where there was less technological or cultural means of assuring paternity, but we don't really know. That matters because the frequency of non-paternity events in prehistory, when our instincts for handling the possibility of non-paternity would have evolved, would decide how much effort were willing to dedicate to raising our partner's children.
If we assume your herms had a human-like mating strategy then I'd say a 3-6% frequency of non-paternity would be common in prehistory, which means that on an instinctual level a herm is willing to dedicate 3-6% more effort into raising a child it birthed over a child it's partner birthed.
Arguably in the 'traditional' nuclear family, where two herms raise a pack of baby herms, the issue with infidelity is small enough to likely only play a small role in differences in how children are raised. However, there are situations where paternity is less certain where that lack of confidence would lead to more significant preference for children the herm birthed itself.
Studies show that non-paternity events are more common in couples that aren't married, couples with less education, couples of lower socioeconomic level, couples that regularly fight or have marital/relationship problems, and couples where travel or work keeps the two individuals apart for large lengths of time. In these couples the higher chance of a non-paternity event would translate to an instinctual preference for the child one birthed being much higher. As an extreme example if a herm has a documented history of cheating then it's quite likely that their partner is going to favor the child that they birthed because of the very real difficulty of being certain of the paternity of the other child. The point being that there will be certain situations where you may expect to see a marked preference for one child over another due to paternity concerns.
Socially I imagine this would lead to a likely cases of inheritance putting preference on children birthed by a herm over those fathered by the herm. Of course without separate sexes passing inheritance down the male line no longer makes sense, so in most cases, where a couple is married and have intermingled their assets, it wouldn't really come up since neither of the adult herms would have a higher claim to those commingled assets then the other herm does. Still in cases of royalty or other situations where one herm clearly has a greater claim to assets or some genetic lineage then the other herm likely children birthed by that herm would get priority for inheritance and similar policies.
Milking pregnancy for all it's worth
(I really need to come up with a better pun)
Assuming that your herms are roughly similar to mammals then that would mean they need to provide milk to their young. While it's quite possible that both herms would be able to produce milk and nurse a young it's possible that the one that birthed the young, and went through all the hormonal cues of pregnancy, would either be the sole provider of milk or the primary one. This actually factors in to the above point, since a herm is more willing to provide resources to a child they know shares their DNA the female would have more incentive to provide milk to the child then the male would, since she knows she is feeding her own child.
If the female either is the only provider of milk, or the primary provider, then this would also render her the primary caregiver, since until very recently it was impossible for the non-lactating male to feed a nursing child. If this happens the 1-2 years of nursing and child-rearing would likely impart an increased bond between mother and child over that of the father, and likely the mother would traditionally be the primary care giver even after the child stopped nursing, as she would have already grown to know the child's needs and personality better while nursing the child. The fact that the mother already is more invested in the care of the child she birthed, if only by a small amount, would further encourage the mother to continue as child-rearing everything else being equal.
In a modern society where there are plenty of means for a non-lactating individual to care for a nursing child these sort of restrictions need not apply, but cultural and social norms would already have been built up over generations during which only the lactating herm could care for the child. These norms would still be taught and passed down to children, even if they don't necessarily make sense in a modern society, and thus would likely still be impacting gender/parenting roles even in a more modern society.
Of course this is one area you get plenty of say in, it's easy enough to create herms where either both partner's lactate equally or children don't nurse, in which case this entire section becomes moot.
It's allot harder to be a dead beat mom
One could expect that issues with a male not sticking around to care for the child will still happen in a herm society just like it does in our current society. Thus single mothers raising children will happen far more then single fathers still, and thus the legal system will likely have protections in place for herms who birthed a child if her partner isn't assisting in providing for it. In fact if a herm is raising a child by itself it will likely be presumed to have birthed the child for this reason.
Gender disparity will still exist, because you can be a dick even if you also have a vagina
I need to add a very important caveat to this entire section, that I'm speaking purely about evolutionary psychology. The below is a true and known aspect of evolution that can be demonstrated. That does not necessarily mean I believe that it should be used to dictate anything about how modern society should be structured or believe it makes a significant difference between sexes in modern society. In fact I'll go so far as to say someone trying to use this as claims of superiority of one sex over the other or to justify treating sexes differently in modern society is butchering evolution and science and is all around a terrible person
It's a known fact that herms have a preference when it comes to mating, they would prefer to play the role of a male. Males dedicate far fewer resources into producing young then the female does, allowing the male to dedicate those extra resources in potentially mating with other herms and producing more offspring. This preference is more significant with a polygamous mating system then a monogamous one, as the male is free to mate with other herms immediately after impregnating a female in a polygamous mating system. However, even in a monogamous mating system this preference will still exist as there is always sexual conflict with the two sexes trying to offload as much work as possible on their partner's even in monogamous systems.
Thus the evolutionary psychology of the herms will be two fold, first a preference for being the male, and secondly a belief that whichever herm played the role of the male is likely 'stronger' or more evolutionarily fit then the one playing the role of the female, as he was able to somehow convince/force the female to accept the less preferable role.
This bit of evolutionary psychology is going to have a significant affect throughout all of your society and culture, which will generalize favor the ideal of herm-as-male. Your see it in everything from royalty and other rich/important figures always insisting on mating as the male to society treating unwed females as a bigger problem then unwed males to Axe body spray commercials promising that your get all the herms begging you to be their male because all the herms are culturally encouraged to be males over females on some subtle level so playing the role of male is just 'sexier'.
Unfortunately we have also seen consistently that those with power will do everything they can to consolidate power, in fact males in human history did quite a bit of power consolidating by exploiting a small physical size advantage into things such as laws and policies that gave females less legal rights as males to policies design to refuse women education or the ability to get a job (so they had to stay dependent on their male partners) or spreading of misinformation about women that denigrated them to make males seem more powerful by comparison (like claims that women are less intelligent or that they were too 'emotional' to hold a job or that they were more comfortable being subservient).
Please note this is not a situation limited to males. Females have been shown to be just as willing to consolidate power whenever they were in a position of power, it just so happened that differences of physical size and childbearing costs generally put males in a better position to do so in the past.
In a herm society I could see somewhat similar tendencies occurring. While any herm could mate in any position the ones in power, be it financially, politically, or physically, will both consolidate power, to put themselves in a better position in the future, and likely insist on mating in the male role. As a side effect they may also create systems that make it harder for a herm who has already mated in the female role to gain 'power', in any format, in order to cut back on competition and ensure that once they have mated as a male they will get to keep that position.
As an example I could see herms making claims that females who are pregnant or nursing shouldn't be allowed to attend school as they need to dedicate all their energy to their child and/or would be too distracted/distracting to get her education. While there would be all kind of arguments why this just 'made sense' one of the unspoken reasons would be that of consolidating power. If two herms mated at a young age the one that mated as a male would be able to continue going to college while his partner couldn't, thus he would have the education that allowed him to get a better job and thus be better suited as the 'bread-winner' then the female who was refused an education. When the time comes around for another child to be had he can thus argue he is too busy bringing in the money to be pregnant and thus insist the other herm again be the female. Basically taking one opportunity of being the male in the first mating as an excuse to justify his gaining the skills necessary to always get to maintain the role of male.
All the above is talking in very large sweeping terms across all of society. It's quite possible that many herms would regularly swap the male and female roles and that some herms would have a preference for female role for instance. Likewise it's quite possible many of the herms who created or propagated systems which helped males consolidate power may not fully understand how those systems were in fact assisting in the consolidation of power. In short, I can anticipate patterns likely to show up across the society as a whole, but that doesn't mean you should presume every individual in that society share's that belief or motivation.
Regardless the net result is that a sort of gender role where herms mating as males were preferentially favored, or herms that mated as females were somehow subjugated, is not at all impossible. If such a case existed that would then have a very significant effect on how a child would treat each of their parents. For instance such a system would likely encourage forcing female mated herms to be responsible for child rearing, which in term would affect the parental bond with the child as in my previous point. For that matter the child may grow up to treat each parent differently based off how society treated the two types of mating. For instances your likely to hear "my dad can beat up your dad" sort of arguments which cite the herm that mated as a male as the one to beat up because society expects male mated herm's to be 'stronger' then female mating herms.
Put more generally once society creates gender roles based off of method of mating those gender roles are going to affect all aspects of life, including how the children and parents interact with each other. A whole separate question could be dedicated to just how many difference such roles could create, so I'm not going to waste time explaining each one in depth.
Down with gender roles
If you don't want these sort of gender roles to be as prominent in your world there is, luckily, a simple evolutionary justification for decreasing these roles. If monogamous mating are the norm then it's possible that herms regularly alternated between the male and female mating role every other mating.
From an evolutionary stand point this would make sense because, as I stated before, it takes more energy for the female to birth a child then for the male to father it. If the goal is to pop out children quickly then waiting for the female to regain the lost calories/resources she expanded in birthing a child could slow the rate at which pregnancies could happen. By alternating who plays the role of male it would be possible to better spread out the resource cost of multiple mating's, thus potentially allowing more total mating for a monogamous pair.
If such a system had evolved before modern society and culture evolved then gender roles would be far less drastic, since it would be the evolutionary norm for partners to alternate matings and thus every herm would be expected to occasionally mate as a female. This wouldn't entirely remove the problem though, as evolution would still favor the male role of mating and that would still have some affect on their evolutionary psychology. In particulate there is a possibility that modern societies, where medicine and technology made child mortality lower and thus couples spent more time on raising existing children rather then birthing new ones to replace deceased kids, that a preference for the male role may grow in 'vogue' as it no longer became necessary to alternate roles in mating just to keep up with the demands of constant pregnancies. Still, this would at least significantly decrease the degree of preference for specific gender roles.
There is another catch with this possibility, the fact that it only makes sense when your goal is to constantly produce kids. Putting most of your effort into producing lots of kids, rather then in raising existing kids, is more of an R-type strategy by definition, and generally sapience is primarily a k-type mating strategy. In layman's terms the really smart animals likely would have a large gap between pregnancies while the took the time to raise and teach their newborn smart child how to use his smarts to survive, and if that happened then a herm has enough time to catch up on the calories lost during the birth in the years between pregnancies spent raising the existing child.
The solution to such a problem would be to put the herms in a harsh environment where infant mortality is extremely high. I'd cite penguins as the perfect example of this. The male and females trade off time spent caring for their unborn child (in this case an egg) by passing the egg back and forth because it's the only way the two of them can manage to raise an egg in such a harsh environment. While alternating pregnancies is a bit different then trading eggs there are quite a few analogs that one can draw. still I'm getting a bit off topic now so I'm going to stop here by saying if you wanted to ensure herms traded off mating roles it's probably worth asking a separate question so we can go into more detail about the reasons.
Who's your daddy?
Finally there could be quite a significant difference in parentage based off of mating role because it's entirely possible that 'realistic' herms would never know their father. Generally hermaphroditism only persists in a world where it's hard to find mates due to individuals of the species being highly isolated. I already went into quite a bit of detail as to why that was at the beginning of this answer, so rather then repeat myself I'll encourage you to check out that answer.
The point being it's quite possible that herms only met up long enough to mate and then went their separate ways, meaning a young herm would only be raised by it's mother and thus there would be a significant difference in how a herm was raised depending on which herm was their mother.
This point gets a little complicated if you actually want intelligent herms with their own culture, because the degree of isolation necessary for hermaphrodites to stay an evolutionary good idea isn't exactly supportive of evolving sapience or culture. In fact I have a separate question just for trying to justify the existence of sapient herms.
However, I can say that it's quite possible that herms at least started as a mostly isolated spaces and thus their instincts are less in favor of monogamy and/or put much higher emphasis on the maternal bond over the paternal since in their distant ancestry the paternal bond didn't really exist, how much that is true depends on how you justified your sapient herms existing in the first place.
