Despite the obviously evolved intelligence of the chimpanzees, the point is made -- twice -- that chimp leaders ascend by virtue of sheer physical power. There are two fights between Caesar, the good chimp, and Koba, the evil one, to determine who will lead. Both times the other apes seem content to passively await the outcome.
My son asked the perfectly logical question: if might rules, why aren't the gorillas in charge? (The gorillas in the ape colony tend to fulfill more blue collar roles, sentries and the like.)
But it's hard not to think that some of this was also an outgrowth of some recent real life chimp attacks, the most publicized of which involved that unfortunate Connecticut woman who was mangled by her friend's pet chimp.
In Africa, the locals know full well that chimps, when given the opportunity, will kill and eat young humans. Chimps tend to bite off the face, fingers, and genitals of humans first, so their attacks are particularly gruesome.
Gorillas, on the other hand, kill rarely. Of course, there are far fewer gorillas than chimps, but even given that, gorillas, despite their more intimidating appearance, are simply less vicious. When confronted by a human, they are far more likely to bluff charge, or beat their chests, than actually attack. This is as befits a herbivore. (Chimps are omnivores, like us.)
So, maybe the underlying message of the movie is that it is not the most intelligent who rules, but the most vicious and dishonest.
In that sense, they are not so different from humans.