Light skin is mostly a cosmetic trait selected for by some mothers favoring it in their children. It doesn’t seem likely that Neanderthals were that deep into cosmetic selections. Dark skinned Neanderthals must be politically incorrect to depict.
In northern climates, you need light skin in order to pass enough sunlight to produce enough vitamin D. It's possible that the genes for light skin came from Neanderthals breeding with humans rather than evolving in the rather short period of a few tens of thousands of years.
Light skin was favored by natural selection in northern climes. Sun strikes skin, forms Vitamin D, which is absorbed and causes better pelvic development for child bearing. It is quite likely that Neanderthals were predominantly fair skinned, blue eyed, redheads. Homo sapiens out of Africa did not do well in the far north until the white gene mutation occurred, perhaps 30 or 40,000 years ago, allowing for more successful childbirth.