"While 20% of the Neanderthal genome survives today, most people only carry about a few percentage points of Neanderthal DNA, and most Neanderthal-derived DNA is non-coding. Neanderthals maintained a low genetic diversity and suffered from inbreeding depression; consequently most Neanderthal genes were probably selected out of the gene pool...
"Various estimates exist for the proportion, such as 1–4% or 3.4–7.9% in modern Eurasians, or 1.8–2.4% in modern Europeans and 2.3–2.6% in modern East Asians...
"Neanderthal [mitochondrial] DNA (which is passed on from mother to child) is absent in modern humans. This is evidence that interbreeding occurred mainly between Neanderthal males and modern human females.
"There is considerably less Neanderthal ancestry on the X-chromosome... The lack of Neanderthal-derived Y-chromosomes in modern humans (which is passed on from father to son), has also inspired the suggestions that the hybrids that contributed ancestry to modern populations were predominantly females, or that the Neanderthal Y-chromosome was not compatible with modern humans and became extinct."
Also he's a pretty good actor.