I wasn't trying to propose any specific theory about the demise of Neanderthals, but if I had to, it would be this:
There are some very persuasive arguments being proposed by numerous anthropologist/paleontologists for human self-domestication. Those conclusions are based on a constellation of traits that domesticated species all share to some extent -- both morphological, developmental and psychological. If you're curious, follow the topic up for all the particulars.
If humans did indeed go through this domestication process (which I tend to believe), but Neanderthals did not (they don't share the traits I referenced above), then humans had an edge over them in terms of the degree of cooperation you could reach in a group of individuals. I don't equate that with "sociability" per se, but maybe that's quibbling over terminology. But larger human groups could cooperate in hunting ventures or other activities in which there is strength in numbers, and that could have conveyed more resilience than the Neanderthals experience in bad times. Even a small edge can make a big difference during the thousands of years in which the two species co-inhabited the same regions.
This doesn't rule out both co-mingling genes and outright fighting. Groups can do both over the course of centuries. As Wrangham outlines in his book, one of the common dynamics in premeditated violence between groups (whether chimpanzees or human hunter-gatherer cultures) is whether or not there is parity in numbers. Chimps won't attack another individual or group unless they are overwhelmingly in the majority; a typical ratio is 8:1. This is the number of chimps it takes to hold a single individual down and kill them, without a high risk of being injured in return. So when a patrolling group of chimps encounters a lone individual from another neighboring troop, if the ratio is less than 8:1, they won't attack, but if they've got the numbers, they will murder that lone chimp.
According to Wrangham and his sources, hunter-gatherer groups showed this same sort of dynamic. Within their own tribe/group, reactive aggression was almost non-existent; they were very peaceful among themselves. But if they had an advantage in numbers over a neighboring tribe, they would attack. It had to be a significant advantage, however, because the goal was to inflict damage without a high risk of injury.
If we extrapolate that same dynamic to early humans and their interactions with Neanderthals, in times of numeric parity, there were probably peaceful relations. But if at any time humans had an advantage (and possibly vice versa), then there was an increased risk of violence between the two species (sub-species, whatever).
I'm sure there were many factors in the decline and eventual extinction of Neanderthals. I very much doubt any one single factor could bring them down. But it wouldn't surprise me if humans weathered changing climate and other pressures better, as a group, because we had a greater ability to cooperate with other humans, and when we had the numerical advantage over another group, we attacked them and took their land and resources. It's a pretty deadly combination.
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