Instead of trying to prejudge the answer by listing what sorts of behaviors I had in mind, I’ll leave it open. In what ways do humans behave like apes?
I think your question is too broad. Why not get at exactly what you’re trying to figure out? Also, do you mean “great apes” or just “apes”. Or, do you mean chimps/bonobos. If the latter:
For instance, we both scratch ourselves when we itch, we both eat when we’re hungry, we both have sex, we both suckle our young, we both die of old age, we both eat meat, we both form hierarchies in groups, we both play, we both use physical violence to obtain “things”. One could go on all day…
This is not a behavior.
In pretty much all of them, except for those behaviors which are distinctively human. It would be better to ask which behaviors are different in humans than in apes, rather than to ask which are similar.
Even within the great apes, there is a great diversity of social structure, ranging from solitary in orangutans to male-dominated harems in gorillas to female dominated groups in bonobos. So many of the large variety of social structures used by humans can be found in one or another of the great apes.
As others have noted, there are relatively few different “basic” behaviors. The behaviors that are (seemingly) “different” are primarily distinguished by the contextual complexity of our substantially more sophisticated use of language and symbolic processing.
Because of this humans and apes exist in very different mental contexts, even if our primal needs are quite similar.
Read “The Third Chimpanzee” by Jared Diamond, in which he attempts to prove that all of our characteristically human behaviors have evolutionary analogues among other chimpanzee species. (Oh, he half-seriously suggests we should classify humans as the third chimpanzee species, hence the name)
I just watched an episode of “Dirty jobs” and they were throwing poop at each other.
Does that count?
When humans pick nits, it’s in an academic sense instead of a physical one. 
One difference I can think of is the human desire to remove adhering feces from the anus after defecating. We used leaves and water and other natural materials before (and, in some regions, after) the invention of paper, but I don’t ever hear of our primate kin doing the same. Is this something that universally bothers humans but doesn’t greatly affect nonhuman primates?
Humans and chimps both wear clothes, but chimps are much funnier when they do it.
These are the sorts of areas I had in mind. Social behavior. Patterns of domination and distribution of resources, formation of social groups, etc.
Eye contact. When I visited Gorillas at the zoo, they sat around each other, but would not make eye contact with the visiting people. I think all those people are stressful for them.
This has been discussed in various threads now and then, but I can’t find the particular one that I followed a few months back, and I don’t have a cite. It’s likey because of our larger glutes that have arisen due to our upright walking posture. The anus of other primates is more flat, with less tissue around for things to get stuck on while doing their business. We can minimize the effect by squatting on our haunches (which many of us rarely do anymore) but we have the added incentive to clean ourselves up back there because not doing so can cause discomfort and hygiene problems when our glutes close together once we stand up and walk.
I’ll also throw out a WAG and say that the scent of not being pristinely clean may have use in the social lives of our primate bretheren, contributing to an individual’s unique smell. We all have anal glands. We might’ve lost the ability to distinguish such things with our (assumedly more limited) sense of smell, making it less likely to be a conserved behavior. But again, no cite.
I’ll throw out another WAG and suggest that , given the diet of most non-human higher primates, their feces usually don’t stick to them.
We both pick nits 
Heck, read The Naked Ape, The Human Zoo, and other books by Desmond Morris.
While there are a number of things I dislike about the man (his lack of footnotes and references in his earlierr works, especially), I find that I agree with much of what he says. And – please note – he edited a book on primate behavior long before he wrote TNA. The man does know whereof he speaks, and he wasn’t the only one to say so. He’s just the only one until recently who got a bestseller out of it.
Human are apes. Apes are simians. Ergo, all human behavior is also simian behavior.
Yes, but the monkeys eat them.
Apes aren’t monkeys.
(That’s for pointing out that I included a non-behavior in my list. 
)
[Stewie] Blast!! [/Stewie]
(I knew that, but “monkeys” is funnier. ;))
When I was in college I was doing a study on Ritalin and Adolescent Development. I was in a pre-school studying children from ages 5-8 some with developmental disabilities others without. There was one young man who when aggitated would walk/crawl whilst on his knuckles. The exact behaviour of large primates in the wild. He was so accustomed to doing this he had a rather large callous on the top of his fingers. When I confronted the mother and asked about the behaviour she said he’s been doing it since he could walk. I’m saying that I’ve never seent his behavior before or since, it was truly ape-like…