fwiw, i think it's pretty much settled that homo sapiens probably just outbred neanderthals, and overwhelmed them with sheer numbers. there was substantive neanderthal introgression into the anatomically modern homo sapiens, but in the end they just got back-crossed out.

one would expect that a species like neanderthal, that was adapted to a cold climate with scare resources, would develop lower breeding rates than a species like sapiens, which lived in warm climates and had abundant predators. when the climate warmed up, allowing sapiens to move north, they would have carried those higher fertility rates north, and there you go.

they tell us we have some percentage of neanderthal genes depending on our existing racial genotypes, but it's never really discussed what those genes are. i would expect that we'll eventually learn that the genes we inherited from neanderthals largely have to do with adaptations to colder climates. if true, the introgression no doubt increased the fitness of the donor species.

https://www.washington.edu/news/2003/09/22/bones-from-french-cave-show-neanderthals-cro-magnon-hunted-same-prey/