Thank you for your excellent question. Assuming that your evidence about Neanderthals is approximately right, I think that you are asking not so much the question whether they are human--that tends to be understood as the question whether they are members of the species Homo sapiens, which they are not--but rather whether they are *persons*. Some people will indeed put forth the use of natural language (like Hopi, Swahili, or French) as the defining characteristic for being a person. However, that is problematic because, for instance, many autistic individuals are incapable of language--yet most would count them as persons nonetheless. By the same token, the Great Apes Project holds that non-human great apes, such as gorillas, chimpanzees, bonobos, and orangutans, should be accorded the status of persons in spite of their not having a language. (By the way, having a system of communication is not enough to have a language. See Anderson's book, _Dr. Doolittle's Delusion_ for a detailed defense of this point.)
So language is problematic as a criterion of being a person. Other suggestions include: the capacity to recognize the existence of other minds than one's own. Your remark about empathy suggests that Neanderthals might have been capabable of empathy, which is often construed as being able to adopt another's psychological point of view. On that criterion, Neanderthals might do fairly well. Yet another suggestion would be that to be a person, it is enough that one be able to reflect on one's own psychological make-up or personality and resolve to modify it in some way. I might, for instance, be frustrated with my short temper, and take steps to control it. This could cause my temper to mellow over time. This "exercising control over one's will" conception of freedom is popular among some philosophers. It is hard to know how we might find evidence that it existed among Neanderthals.
I've laid out a couple of options for thinking about what would show another hominid species to be a species of persons. Less theoretically but more intuitively, I might also note that if we were to come across a heretofore undiscovered group of Neanderthals--living in a remote Indonesian island, perhaps--there might be some debate as to what (if anything) to do with them. Probably we could overcome them, and take them into captivity if we chose. Should we put them on display for everyone to see, for instance in a zoo? Some might advocate this. Others might say that this would be inappropriate, and in particular, morally wrong. Those who think this probably would also be holding that Neanderthals are persons, and not just hominids.