The argument for human uniqueness is of mostly historical interest. As we gained more understanding of animal behavior, we learned that their psychology has more in common with us than had been imagined previously.
Here, then, are some of the criteria put forward for human uniqueness and contradictory evidence.
The (rational) soul Not scientifically observable and therefore irrelevant
Problem solving Kohler's banana-and-box experiments claimed insightful problem solving in chimpanzees
Consciousness Self-awareness demonstrated in chimpanzees, orangutans, bonobos, dolphins, even magpies, using mirror test
Language Initially contradicted by ape language learning that turned out to be methodologically flawed; dance 'language' of bees; birds mimicking human speech; birdsong; complex vocal communication of whales and dolphins
Social learning Widespread throughout animals with backbones (vertebrates); rodents learn what to eat by observing their mothers; cats teach young how to hunt; many birds learn song from their fathers
Tool making Chimpanzees fashion tools used to fish for termites or ants that they eat; flaked stone tools (e. g., Gona Ethiopia, 2.6 MYA) may precede humans
Traditions Birdsong dialects in local areas; rats diving for shellfish in Po River; novel cliff-nesting by Mauritius kestrels; varied local tool traditions in chimpanzees, etc
Artistic expression Painting by apes, and elephants; birdsong?; song of the humpback whale? etc.
Complex technology For most of human history, our stone tools were little better than those used by chimpanzees (i. e., up to 100,000 years ago)
Agriculture Ants maintain fungus gardens; ants herd aphids and 'milk' their sweet secretions
Engineering/building Beaver dams; ant bridges made of own bodies; honeycombs; weaver bird nests; ant lion traps; spider webs, etc.
Environment change/destruction Elephants flatten trees, opening up habitat for numerous plants and animals; beavers flood river valleys, causing more ecological disruption than subsistence humans before human-caused extinctions about 40,000 years ago
Moore's Law Tendency for technological change to increase exponentially over time but this applies only in post-literate societies: Achulean stone tools changed little over 1.6 million years
Linguists would mostly question whether other species have true language requiring an arbitrary association between sounds and meanings (semantics) and syntax (where the order of the words affects the meaning conveyed). Some animal behaviorists disagree. Irene Pepperberg concluded that her gray parrot, Alex, had mastered syntax as well as semantics. Many birds mimic human speech well but may not understand what they are saying.
Linguistic criteria may be biased and other species have communication features lacking in human language. For instance, every humpbacked whale has its own signature sound whereas humans share their names with others.
Finally, it may be premature to deny semantics or syntax in animal communications given that we have a very limited understanding of what they are saying to each other.
So far as the artistic sensibilities of other species are concerned, my favorite proof of this is the fact that art experts could not distinguish paintings by chimpanzees from the work of abstract expressionists like Kandinsky. In recent decades, elephants have come to the fore as painters who even have their own art schools, although their efforts seem more a circus trick than art.
Humans of the last 100,000 years or so are very different from our more remote ancestors so comparing humans to other species is a moving target. One could argue that the key difference between recent humans and all other animals is the accelerating pace of change associated with modern technology (Moore's law). Writing greatly increased the information available to humans and the digital age put this trend on steroids.