From all I’ve read about Neanderthals I can guess why they died off - they were family based and lived in small family units so always found themselves outnumbered by hordes of Homos. I’d even postulate that we are the lesser species and would still be in the iron age if some crossbreeding with the superior Neanderthal line hadn’t given us a needed boost.
My guess is we interbred to the extent that we became one.
I read an article that said Neanderthals were smarter than the early Homo Sapiens but lacked the skills of making and using of weapons. If they were smarter I would have thought they would be able to adapt.