"Competing" is an euphemism. Human evolution is driven by tribal warfare. We didn't out-hunt the Neanderthals, we killed them off. Mainstream anthroplogists avoid that subject like the plague.
There is little evidence the Neanderthals were religious, while the modern humans were. That doomed the Neanderthals because religion provides a significant competitive advantage in tribal warfare. Mainstream anthropologists have selective hearing on that subject also.
Neandertal didn’t go extinct, Neandertal has over a billion living ancestors.