Recently at a human evolution conference in England Svante Paabo (or someone in his group) was alluding to discovering how modern humans and Neanderthals differed by looking at the ~30,000 genetic positions (bases) where modern humans and Neanderthals exhibit fixed differences. That is, Neanderthals and modern humans exhibit totally disjoint frequencies.
I’ve been saying this for years, but I’ll say it again: this is probably a fool’s errand. I do think there are major differences at loci which we know about, such as at FOXP2. But, it isn’t clear that even at FOX2 Neanderthals and modern humans exhibited complete lineage sorting. That is, there’s evidence that the Altai Neanderthal had introgression from modern (or modern-related) human populations, and that those variants were sweeping. And there is still variation in modern human populations at FOXP2.
It looks like Soft Sweeps Are the Dominant Mode of Adaptation in the Human Genome (also see Detecting polygenic adaptation in admixture graphs). But it may be that soft selections and recurring gene flow are common features of the broader Homo lineage for several million years.
In other words, looking for silver bullet variants which can explain why we are so special may always fail, because there are no silver bullets (for several years at ASHG I note that there were presentations which attempted to determine the locus of humanity by looking at the loci of functional interest where Neanderthals and modern humans differed). Rather, human exceptionalism is no exceptionalism, and human populations explore a wide space of phenotypes defined by a huge range of allelic variance which spans many of our extant lineages.