The essential difference is that animals strive to be bored all day while humans have to strive to endure boredom.
In reality, though, it is a matter of degree. Some non-hominid species can be taught basic math, but no other animal could ever understand the mathematics of something like Relativity much less conceive of it. In speech, there is some intense complexity that is hard or impossible to find in other animals. For example, meerkats and many birds can make very distinct sounds based on threats. Essentially saying something like "there is a cat over there" that communicates the type of threat and its location or direction.
However, a human can go a level above in this. If one person said, there is a lion over there, another person could say "Ronnie says that there is a lion over there" while other animals could only repeat the original sound or vocalization without applying the referential subject predicate piece of information.
Nevertheless, these are natural abilities humans are born with that hardly represent the distinct elements of human beings such as the various adaptive organizational developments that truly set humans apart from any other animal. Chimpanzees, for example, develop extremely complex social groups, but take ten chimpanzee strangers and put them in an enclosed space for an hour, and you'll end up with a room full of dead chimps. However, every day hundreds of strangers in every city on the planet board plane after plane for hours upon hours of flights with hardly any incident. The ability to repress natural aggressive impulses to that degree is not seen among even our closest surviving relatives.