For a very good treatment of this subject by a highly respected primatologist, see Frans de Waal's recent books. But he isn't alone... until fairly recently it was taboo to say that animals are "intelligent" or "conscious" or "sentient", but this is thankfully beginning to change, and among the current generation of biologists, ethologists and animal cognitive psychologists many consider the notion that animals aren't sentient to be outmoded, or as Frans de Waal puts it, "anthropodenial".
But there is one thing that makes us truly different from all other animals and it's only indirectly related to cognition: for better or for worse we are the only animal that uses and controls fire. It's fire that really defines humans, not our big brains (and fire probably came first; cooking made our calorically expensive large brains possible). And it is fire that may be the end of us, as well.
Just to nitpick on that one, some wild animals seem to use fire to their advantage, such as birds of prey spreading wildfires. Some others were trained to be able to cook food, and light fire with tools. That said, to my knowledge, none are able to start fire with tools they crafted (I am not really either, to be fair). So this might be true for now, but I do not take it as a given.
I like your last paragraph, though, especially the ending. Nicely put :)
Interestingly, we're starting to find evidence for grammatical rules in other animal languages: https://www.newscientist.com/article/dn20615-first-evidence-...