I just watched the 2015 documentary, Neanderthal Apocalypse. It basically presents the view that the Neanderthals went extinct due to a huge volcanic explosion in what is now Italy. The volcanic ash and resulting climate change caused areas populated by Neanderthals like in Europe and the Middle East to suffer sudden changes. That resulted in their eventual extinction. Homo Sapiens were more spread out into Asia, Oceania and Africa which were not as badly affected and Homo Sapiens had the numbers to survive the natural calamity.
In short. We did not kill of the Neanderthals. Mother Nature did.
I'm waving my fist at you Mother Nature.... How could you?