John Hawks sifts the narrative, and basically decides that species is a semi-useful term that can be largely ignored -- particularly in the fossil records. His read is that Neandertals are probably members of the same species as us, but that the distinction is pedantic and misleading. He does strongly feel that our genome contains evidence of Neandertal descent, but that the discussion is missing the point (emphases mine):
... I think that evidence of introgression reinforces the hypothesis that modern humans emerged in an adaptive context, making use of adaptive variation from a widespread (possibly pan-Old-World) population of archaic Homo. It's one of the two main patterns in the evolution of modern humans.We're all waiting to learn what the other pattern is, Hawks won't tell us yet. The fundamental shift is to stop thinking of Neadertals mating with "us", and to instead think of modern humans as something that evolved from a large number of "archaic Homo" populations. We are Neandertal, but we are also one of many old populations.