I bet it at least ended that way.

They died out during the last ice age when one of the big differences from previous ones they survived was the presence of homo sapiens.

I'm guessing it was from a combination of lack of space, raids, and integration (willing or not).

We (who carry it) have a small amount of Neanderthal DNA, but it's not the same DNA for all of us, it's pretty diverse, so cross breeding events between us weren't limited to just a few times.