I think there are two supporting theories on this. I think I believe the genocide one. They were different so we attacked them. The theory on us mating with them until they became part of the species just seems wrong, I know humans we attack what we don't understand, and that is the case here.
Whether we out bred him (or bred in with him till he disappeared as some more recent theories suggest), or simply pushed him out of his habitat, I suspect Homo Sapiens had quite a bit to do with his demise. Apex predators tend not to tolerate competition, humans even less so than most, we barely tolerate species that actually benefit us, if they can be factory farmed.
My entire answer will be based upon the theory of human evolution and dispersal- this poll is based on that theory but we must remember it is still only a theory.
Homo neanderthalensis (H.N) originated out of Africa about 500kya. They migrated to Europe where the ice age climate created a selection pressure to make them short and stocky (min. SA:V ratio) to minimise heat loss and they had a diet that was 90% carnivorous, so they hunted the large herbivores of the time. Some time later, Homo sapiens (H.S) also diverged in Africa and had a tall slender body (max SA:V ratio) to maximise heat loss- Africa is a hot place. The H.S migrated out of Africa as well, following in H.N's footprints to Europe. Despite not being adapted to the cold climate, H.S was much more resourceful and creative than H.N, being able to communicate (this is under debate) more than H.N. This enabled them to adapt better survival techniques, hunting methods, and weaponry. Using this, H.S hunted most of the large herbivores to extinction, effectively wiping out H.N's main food source. Because H.S was omnivorous, the lack of meat sources weren't a big issue as they could still eat vegetation, but H.N had become specialists and couldn't adapt. It is suspected that many H.N died of starvation, and those who were left were sucked into an extinction vortex.
This is Gause's Principle- no two species can occupy the same ecological niche for an indefinite time without one out-competing the other.
H.N was out competed by H.S because H.S had a better survival advantage and H.N was unable to adapt to match them. Therefore, despite the inbreeding that did occur (about 4-5% gene flow), the Neanderthals did die out whereas we, the Homo sapiens, continued to survive.
Personally, I don't believe the Australopithecus to Homo sapien theory. I don't believe we shared a common ancestor with chimps. There is little to no evidence for it- the fossil record is a joke- but this is the accepted story I was taught in school and therefore I'll give it to you so you know the evolutionary side of it.
Human beings did cause the extinction of the Neanderthal and, yes, this was partially done through war. But, contrary to popular belief, the neanderthal really isn't extinct. We interbred with them, and they died off. Modern humans, with the exception of black Africans, have Neanderthal DNA in them. And quite a bit.
While humans have helped cause the extinction of many different plants and animals, I do not believe they had a hand in the extinction of the neanderthals. I believe that some sort of ice age had a huge hand in destroying their species, for once it seems like the humans may be innocent.
I don't think we can ever know that, but I really don't think humans were sufficiently advanced at that time to be able to cause the extinction of the Neanderthals. Humans might have a small part to play, but I don't believe that humans are the primary cause of their demise.