I'm going to go out on a limb here and say
Not if the hermaphrodites do it right
TL:DR: Don't assume that just because women can breed that they will try to.
Many other answers here are applying modern (or at the least, human) sociological values to an otherwise alien society. If this sexually hermaphroditic society develops a different approach to gender roles (or, more likely, the entire concept of gender) and adopts a more arbitrary caste based system, then you will quickly see similar roles to men/women developing but split along caste lines.
Strapping young girl? Military caste. Go forth, try your best to kill or be killed. Take risks. Don't breed, as death in childbirth is the worst shame a warrior can know. If you grow old enough to become a man then you have an awful lot of experience and can be used to train the raw recruits, not to mention stud the next generation of warriors into existence.
Likely to get killed in the first skirmish? Don't worry: There's always a need for the working caste. Feel free to breed if you want to, but don't expect any special treatment from the other castes. You're here to primarily support the other castes, including taking care of other people's children if required. If you grow old enough to become a man then bully for you.
Wide hips? Good pedigree? Mother has a history of twins? Breeder caste. You get food, protection and your pick of the men. You're likely going to die giving birth, but hey, at least you perpetuated the species. If you don't die and manage to live long enough to be a man then clearly you're good at this whole 'giving birth' thing, and the next generation could use some of your genetic material. Go forth and multiply.
This system requires more social interaction than the human system, and as such tribes can be expected to be more closely knit. If the death rates are similar to those of the human's you're trying to outcompete then you should come out ahead of the game (or at least drawing) by having specialised roles that are not gender specific, and as you'll likely develop specialties pretty quickly (your 'alphas' will likely be female, your wise men will be just that, you won't have any arbitrary traits defining social structure from the get go) your social structure may well pip humanity to the post and make up for any breeding disparity.
Plus it would make for a social structure that is simultaneously incredibly PC and hugely discriminatory.
Disclaimer: I mostly wrote this answer to be contrary. I have no idea if this would occur and, in fact, no way to simulate the social evolution of a pre-historic population, but it's a possible outcome!