Neanderthals were killed off in a neolithic world war which probably lasted several generations, but not thousands or tens of thousands of years. Neanderthals were Europe's apex predator; the main body of them in Europe ate a diet which was over 90% meat including themselves and any other hominids or humans they could catch. Their DNA was roughly halfway between ours and that of a modern ape, their skulls were those of apes with huge eyes for nocturnal hunting, they had the same 6" fur coat as every other ice age animal which is why no needles have ever been found with Neanderthal remains, and this is what they looked like:
Image courtesy of www.themandus.org
Danny Vendramini on youtyube:
http://www.youtube.com/watch?v=mZbmywzGAVs
No living human is related to Neanderthals in any way and the reason for that should be fairly obvious...
There is no evidence that Neanderthals had fur or looked as the picture you posted looks. In fact they stood upright, made tools and lived hundreds of thousands of years before dying out. They have been classified as human, with Human DNA, slightly different than ours. If they were furred that would be evident in their remains. It isn’t. There are Humans on this planet now that share some Neanderthal DNA. I am sure there would have been no mating between them if they looked like the ape you have posted.