It's a touchy subject, because for at least some people, there are some primates we should consider human. Chimpanzees like Nim Chimpsky, and gorillas like Koko, show a lot of human features, and that's not just anthropomorphization. Get to know a great ape, and you'll find that they're a lot closer to human than many people are comfortable thinking.
One key thing they can't do, however, is talk. Language is one of the key things that is distinctive to humans, and even people who don't speak the language well are considered outsiders to your group, if not your species. Koko was able to make a remarkable array of sign language signs, but there's no evidence that she ever mastered grammar even at the level of a five-year-old human. Nim Chimpsky could definitely understand some spoken words, but there's no evidence that he really mastered what we'd call language. Mind you, there are are least some humans who never get to that level, either, and we don't consider them out of the "human" category.
Nobody knows quite what Neanderthals could do, and that would have been important in how people felt about them. There is some evidence that Neanderthals could talk, in the size of their brains and the structures of their bones, but it's far from conclusive.
Without doubt they used stone tools, and created elaborate funeral ceremonies. They cared for each other; there are skeletons of individuals who clearly survived injuries and birth defects that they could not have managed on their own.
Having an actual Neanderthal in front of us would tell us a lot. They were certainly more human than any existing animal, and as I said, there are some primates who may deserve more consideration for human-ness than they do. I might not let a chimp vote, or attend school, but it may be that keeping them in cages is inhuman.
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