John and I have an article out now on Neanderthal introgression: Dynamics of Adaptive Introgression from Archaic to Modern Humans. It's in Paleoanthropology
The major point is that Neanderthals and modern humans were probably interfertile and most likely interbred - and that we would then have picked up most favorable Neanderthal alleles. Which may have something to do with the cultural ' big bang' that happened not long after.

The argument relies on a simple result of population genetics, due to Haldane - that the probability of success of one copy of a new favorable allele is 2s, when s is the selective advantage, much higher than that of a neutral allele.

As far as I know, this conclusion has not been published before.

You have to wonder why.