This may be why coastal Portugal is the last known settlement of Neanderthals.
I think the reason they went extinct is staring us in the face. Homo Sapiens were better at getting food.
Consider:
Homo Sapiens came out of the African plains. They were adept at efficient travel over long distances. Neanderthals were very muscular and inefficient travelling long distances. As populations increased and food grew scarce Homo Sapiens used less calories to acquire food and therefore required less food to maintain themselves.
Homo Sapiens had deeper voices than Neanderthals therefore their voices traveled longer distances than Neanderthals. This provided a much larger area for Homo Sapien hunting groups to operate in using coordinated, self-supporting hunting tactics.
The Neanderthals simply starved to death.
Deeper voices don't travel greater distance, it's quite the opposite. Of course, that quibble isn't the salient point -- the salient point is, the Neandertals didn't go extinct, the Neandertal are among the ancestors of most of the people in Eurasia and a good many outside (thanks to the Great Migration / Age of Sail).