But seriously, primate research is worthwhile on it's own. You can't directly prove connections between us and them; but that is not unlike a huge amount of science which relies upon observation, statistics and expert judgement calls. When they act similar that can be a clue or merely a coincidence but it warrants further investigation. It's a technique that allows for faster probing of the problem space. BESIDES, if you think that observing chimps influences their behavior (as the "skeptics" denier fanatics always claim) just imagine trying to study humans! Humans are way more difficult to study without influencing their behavior thereby tainting the study.
The biological connection is obvious; any "insights" they do in the research with a biological connection become useful even if at 1st they seem unjustified. The work can be applied in new ways later on if one proves there is no connection.
As far as pure human nature study by primate research; well, that is based upon theories which may or may not be proven some day in the future. You have the classic old Feud work on "base desires" which think about the primitive instinctive aspects behind the manifestations; his work in this area is the basis for modern propaganda (WW1 and really big after the Nazis used it so well. Today, it controls most consumers.) Following that success, one could approach further from that perspective - making our relation to primates and their more primitive state ideal.
Ultimately, I think most the work in the area ends up with the search for biological parallels between us. Say that HATE is really just a manifestation of FEAR; can you ever prove such a thing? nope. not in a hard science way; it's all subjective. But if you can study primates you might find more concrete proof with them on a biological level. A Turing machine is a lame computer nothing like your CPU but it's useful to prove things (the difference being that you can concretely prove the CPU is equivalent to the Turing machine and you can't with a chimp since they evolved differently even if they are nearly the same DNA.)
As far as evolutionary pressures-- the best theory for human brains was we already made it to the top of the food chain being as primitive as chimps and what made the apes of the plains smart was that they had to war against each other for resources. Just like humans have always done; my tribe and me against you and your tribe. Given how we are the most evolved distance running animals on earth, territorial borders are meaningless. Your group isn't going to give up after running down a lion to exhaustion for 20miles simply because you ended out of your usual turf.