Neanderthal would quite possibly be more accurately described as a 'race'.
The jury is very much still out on this one, with no sign of clearing up soon. It is a pretty controversial topic in the field of physical anthropology. Science are a long way from consensus or overwhelming evidence one way or the other on whether neanderthal was: 1)essentially a highly adapted and isolated racial/ethnic variation of Homo sapiens; 2)a subspecies --which, at any rate, is not well defined as being different from an isolated racial/ethnic variation-- for example Homo sapiens neanderthalensis, as opposed to modern huans, Homo sapiens sapiens; 3)possibly a separate species. Even if it can legitimately be characterized as a separate species there is, yet again, no consensus, nor overwhelming evidence as to whether they died off or just reassimilated/interbred with archaic modern human populations.
And yes contrary to popular myth some distinct 'species' can, and do, interbread and produce viable offspring. This is due to many factors. Taxonomy is a human classification system for information, and does not necessarily reflect biological realities. Accepted taxonomies are constantly in flux due to availability of new data, as well new interpretations of data. The definition of species itself as not yet stable, and is still somewhat controversial. Some distinct 'species' don't interbreed because they... haven't. This can be due to geographical or even 'social' isolation from other populations, and is not necessarily an indication of biological inability to produce viable offspring. BTW, in evolutionary biology 'viable offspring' means they can go on to produce and rear new generations of offspring who are also viable. The ability to interbread as a defining characteristic for speciation is fairly antiquated. It's a handy item to have at the top of the checklist: if populations biologically can't interbreed then obviously they are different species; it doesn't work the other way around though.
BTW, Tolkiens FICTIONS (as much as i love them) have zero bearing on the realities of evolutionary biology, or social sciences, other than the fact that they are an interesting reflection of the endmic racism of his culture at the time.
Disclaimer: i'm niether an anthropologist nor a biologist by trade nor training.