They didn't disappear. They interbred. White Europeans have Neanderthal genes. Black Africans have no Neanderthal DNA.
Indigenous Peoples Day! Everybody get indigenous!
Actually they do.
If you want to bore yourself to death read this article.
Chad Genetic Diversity Reveals an African History Marked by Multiple Holocene Eurasian Migrations
A quick summary is that .5 percent Neanderthal DNA was found in the groups studied.