This is a recent talk at NIH, which contains some unpublished results.
In the final part of the talk (you can skip there via this link), Paabo discusses the genetic variants (~30k SNPs) that are fixed in essentially all modern humans, but are not present in the Neanderthal genomes sequenced thus far. These variants are presumably responsible for the differences between Neanderthals and moderns. Paabo obviously believes that enhanced cognition is one of the main differences, and he discusses the archaeological evidence for this. He also discusses functional investigations in genetically engineered mice, and advocates for large GWAS that might identify rare humans with "back-mutations" to the Neanderthal variant. Such studies could identify phenotypical effects.
In his recent book, Paabo wrote
(p.213) ... we estimated that the total number of DNA sequence positions at which Neanderthals differed from all humans living today will be on the order of 100,000. This will represent an essentially complete answer to the question of what makes modern humans "modern," ...See also The genetics of humanness, The Neanderthal Problem, and Genetic engineering of monkeys using CRISPR.
(p.253) [last paragraph of the book!] ... One can imagine putting such changes into cell lines, and into mice [or monkeys] ... in order to "humanize" or "neanderthalize" biochemical pathways or intracellular structures ... One day, we may understand what set the replacement crowd [moderns] apart from their archaic contemporaries, and why, of all the primates, modern humans spread to all corners of the world and reshaped, both intentionally and unintentionally, the environment on a global scale ...