Being a social animal is in a class of fascinating things where their greatest strength is their greatest weakness. In the case of being a social animal, the greatest strengths are found in connections between individuals, which let them rise up above what they could do themselves. The greatest weakness of social animals is the connection between individuals, which can pin down and crush an individual who could do better without them.
I like to treat being a social animal as part of a continuum, rather than a discrete jump from asocial to social. I find this is more realistic. It also lets me draw from prior art found in other jumps, such as the jump from single celled to multi-celled organisms. When you really think about it, being a social animal is almost an identical step to becoming multi-celled.
The fundamental weakness of these decisions deals with size. Large things move slower, as a general rule. We spend a lot of time trying to make them move faster, but in practice, a single cell can respond faster than a body can. A single individual can respond faster than entire government can. Almost everything you see in a multicelled organism or in a social animal is trying to combat this general trend.
Consider a punch to the face. Fortunately, for those who are visual, this is the internet, so it's not easy to find examples. If you get punched in the face, you tend to bruise because your blood vessels burst. Now those blood vessels, on their own, were more than capable of rolling with the punch, and swinging out of the way. They didn't need to burst on their own. But, they aren't on their own. They're connected. They're connected to a skull and a brain. This brain has kept the face blood vessels alive, well, and fed, for many years. But now it's a liability, because it's harder to accelerate a whole brain than it is a few blood vessels. And so, under the strain of this connection, they burst.
Obviously we find this to be a beneficial trade. We like our brain protecting our blood vessels, for the most part. But it was certainly a drawback. Those cells couldn't respond to the force of the punch as quickly as they could have on their own.
We see this in cultures too. People are made better by the connections they have to others. We transfer lessons learned, respond to attacks, and do all sorts of useful things with our social connections. But they also slow us down. We can't act out of sync with our environment. To pick on a group that can't defend themselves in these forums (due to social pressures), consider a neo-Nazi who starts to ponder whether this way of life is really the correct way of life. They are surrounded by neo-Nazis and those connections prevent this individual from moving quickly on this topic. It takes time and persistence to leave your social group. Now if we move to more mainstream groups, consider what happens when a conservative starts to ponder the values of the liberals, or a liberal starts to ponder the values of the conservatives. In both cases, their social group will indeed slow them down.
Also related to this weakness is symbols. I would consider the use of symbols to be the greatest strength and greatest weakness that comes from having these social connections. This is language. Its's powerful. It's also very dangerous. It is easy to confuse the symbol for what it symbolizes for.
Consider the human body again. We have symbols throughout the body. The most noticeable are hormones. Consider adrenaline. This is a hormone which basically indicates that it's about to be "go time," because you're going to have to act really quickly. When cells sense this compound, they tend to stop trying to do long term things (like repairs) and start focusing on making sure they can respond in the short term. It's a great hormone. Its part of our body's solution to how slowly large objects move. But it's also easy to mistake the symbol for what it symbolizes. The body has evolved to treat adrenaline as "go time." It doesn't ask questions. If adrenaline hits your heart, your heartrate will go up, regardless of whether you are about to engage in hand to hand combat (which needs the elevated heart rate), or if you're nervous about asking out a girl (for which I really wish the heart rate would stay steady for me!), or if you have a tumor which is causing adrenaline to be produced unnecessarily.
Likewise, our words can be forged. If someone can use words to convey a meaning, they can elicit a response. If that someone doesn't have our best interests in mind, this can be undesirable. This is at the heart of propaganda. Propaganda works because we want the words they use to have meaning, because it's part of how we connect to others. Those words are important to us, and we can't let them go. So the propaganda gets to elicit an emotional response that it probably shouldn't.
For the most part, social animals appear to have an advantage. Or at least we like to think so. Its not all that surprising that we judge animals on metrics that make us look good. We tend not to measure species by things like biomass, because they make other species look superior. (ants monopolize something like 20% of the terrestrial biomass) We don't actually know whether these weaknesses of being social will eventually overwhelm the strengths, or if the strengths will make us prosper. It looks like the evidence suggests its a good evolutionary trait, but evolution plays a very long game. Far longer than it takes for new scientists to grow up and write papers on social animals.