The Neanderthals aren’t gone, they’re still here. And the whiter you are, the more Neanderthal that’s in you. We didn’t out-compete them, we cross-bred with them until their phenotype became indistinguishable from ours.
Exactly. Compare with aboriginal Americans-- full-bloods are rather thin on the ground in the Lower 48, and it's only been 500 years since the first Europeans came ashore in Florida.