What behaviors or actions are uniquely human? Science? Religion? Ethics? War? Certain sexual behaviors?
We don’t really know if any can be classified as such. We can guess as to what other animals do or don’t, but perhaps the same level of behavioral complexity exists in animals as well, it’s just we are unable to comprehend it or even notice it from our context.
We know animals communicate. How well? Anybodies guess. For any given data set there is always a wild extremely unprobable set of circumstances that leads to the same evidence but completely different conclusions. When doing any kind of scientific work this is mostly irrelevant, Occam’s Razor and all that. However, when it comes to comparing humans or human related things with non-humans, then things get tricky, philosophically speaking. Occam’s Razor and the scientific method are human constructs, in fact, it would be very reasonable to assume that anything we can come up with will have a severe bias towards human ways of thinking and reasoning. Measuring this bias is impossible, for we are human and any measuring tool will be equally biased. See what I’m getting at?
Actually, probably the only non-GD answer to your question is “being human”.
The only thing that seems to set humans apart from other animals is the fact that humans tend to seek some kind of criteria that sets them apart from other animals; in all other respects, the difference is not an absolute one, but just one of degree.
We are the only extant species that burries its dead (or otherwise disposes of dead bodies).
I bet there are some social insects that dispose of their dead away from the nest/hive.
I think you’re mistaken, but I’m by no means sufficiently knowledgeable regarding entomology to be certain. Ants immediately come to mind, and I’m pretty sure some species of ants do dispose of bodies.
I see Mangetout has also mentioned insects, while I’ve been slowly checking/writing this post
I think there may be other vertebrate species that do stuff with or about dead bodies (other than carnivorous spp. consuming them), but again, I’m too long away from academia. Certainly elephants pay homage to the remains of other elephants which/whom they knew (which presupposes a sense of smell beyond my imagination, and a memory truly deserving of the adjective elephantine).
It is true that a chimpanzee mother whose infant dies may carry the body around for a time, but that doesn’t count for the present question. Our closest relatives don’t seem to memorialize their dead. Of course, there’s still too much we don’t know. IIRC, Koko (the gorilla who signs) was known to remember and miss her first kitten, "All Ball " for at least 13 years (per an AOL chat in 1998).
You are probably correct.
How about: Humans are the only animals to make and control fire.
That’s probably correct but isn’t it just a specific case of tool use?
Now I don’t know much about animals.
But what about:
-Heterosexual oral sex.
-Playing intellectual games. Like chess.
-Suicide in a way that doesnt hurt anyone else.
Posting to the SDMB.
-Bonobos
-How could we tell?
-My dog killed himself by stopping eating. It’s not uncommon in elderly animals. Is it suicide? Again, how can we tell?
I’m going to go with creating representational art.
Cool. I didnt know about the bonobos.
They would need some sort of pieces to play the game. But yeah, I guess we cant tell.
What about suicide of a young creature, without it hurting anyone else. 
About the art. We have seen monkeys make art. How would you know if its representational?
(If there’s anything sexual a bonobo won’t do, I’d be surprised. Horny, creative monkey sex.)
Have we seen wild monkeys make art, or only ones raised around and encouraged by humans?
Mark Twain said that man is the only animal that blushes, or needs to. He was wrong. Octopi blush when threatened, and several animal blush as part of mating rituals.
We are the only species that makes drinks called Buttery Nipple, Fuzzy Navel, and Sex On The Beach.
Some scholars speculate that it’s not the sense of smell that helps them recognize dead loved ones, but touch and remembering the contours of the face. When elephants greet one another, they tenderly run the sensitive tips of their trunks over one another’s features, much like a blind person feeling out another person’s face with the tips of their fingers. Elephants can recognize others they haven’t seen in decades. (Though to us they “all look the same” elephant faces vary quite a bit.)
When elephants find elephant bones, they do the same thing: tenderly caress the contours of the face. Some researchers speculate that it’s this touch-memory which helps them recognize the deceased rather than smell, since bleached bones wouldn’t give full olfactory cues.
They have been seen to bury elephant bones, by covering them with brush and kicking dust over them. They do not do this with bones of other creatures. They also sometimes carry bones with them for a while (jawbones seem to be a favorite.)
Elephants also love to paint. They will do so with sticks dipped in water, or pieces of fruit that they smear on stones or walls. In the book When Elephants Weep the author reports that a zoo keeper gave elephants paints just to see what would happen, and to her surprise, it did seem that the elephants were trying to be representational. If the elephant saw a yellow truck, it would switch over to using yellow paint, or if a visitor in a red shirt stood by his pen, he would pick red paint. Of course, they lack the motor control to draw as finely as humans, but it appears they may have the desire to do so.
Koko also seems to express sympathy/empathy. One of her trainers lost a baby in late pregnancy. The trainer says Koko had expressed a lot of interest in her bulging tummy, and had shown a delight in feeling the baby kick. When the trainer came back after recovering from the miscarriage, Koko signed “Where baby?” seeing her flat stomach. The trainer signed that her baby had died. Koko immediately replied by patting the grieving woman and signing, “Cry.”
Interestingly enough, Koko has tried to teach other apes sign language, but none of them ever reached the fluency she had, and successive generations slowly lost the ability. I would posit that it’s because, amongst themselves, apes don’t need sign language to effectively communicate. They have a complex system of body language.
Anyone who’s ever had a dog can tell you how attuned animals are to body language. They often pick up on cues we don’t even realize we’re sending. How many times have you heard someone say, “Fluffy didn’t like John Smith, and he turned out to be a real jerk. I never trusted that guy.” Fluffy probably wasn’t picking up on John Smith’s vibes as much as she was reading her suspicious owner’s body language. The owner subconciously told Fluffy to be wary of that man. Likewise, owners who are concerned their dogs will get in fights with others they meet on the street tense up, hold the leash more tightly and accidentally put the dog on alert.
Maybe we humans evolved speech because we were too obtuse to one another’s body language and had to do so in order to communicate. 
They have the fine motor control to deftly pick a single leaf off of a branch with their trunk, but not to control a paintbrush? I find that hard to believe. It seems the lack must lie elsewhere. I’m all for the awesomeness and intelligence of elephants, but there’s no reason to believe they’re not playing with paints the way a two year old does - to explore the colors, textures and even tastes, not to create art for its own sake.
How about this? Humans are the only animals who depend on tools. Many other animals make and use tools, but they can get by without them. Strip a human of all es tools, though, and the first thing e will do is find or make more.
Humans are the only animals with a preserved (written) tradition. At most, other animals only have an oral tradition.