Nobody knows why neanderthals died. There was considerable amount of crossbreeding - modern europeans may carry genes inherited from neanderthals - so it's far from obvious what exactly happened.
The same can be said about native Americans but we know better.
What makes you think pre-agricultural stone-age Homo Sapiens with a sparse population would have been in any position to eradicate Neanderthals to extinction?
Considering modern Europeans all have Neanderthal genes, they could have simply just been bred out of existence.
I imagine all human migrations resulted in genocide - everywhere they happened. This isn't Europe specific.
...it's not even human specific as other animals do the same thing.