I'm not sure it's all that clear cut. These two papers are dense and difficult to absorb, so upon more readings, I may find that I'm incorrect, but I didn't see anything in the paper indicating that we have inherited any of our specific skin lightening genes from Neanderthals.
The only gene mentioned was the BCN2 gene which researchers think is somehow involved in the skin pigmentation pathways. However, they have not yet found its precise function. The best guess seems to be that it may be a precursor gene for the functioning of the color draining alleles.
Human Pigmentation Genes Under Environmental Selection:
http://genomebiology.com/2012/13/9/248#B38
The above paper also addresses many of the issues we've been discussing, including what might have triggered selection for blue eyes, advancing one theory I hadn't hear before, which is that since blue eyes absorb more light, they might have proved helpful in locations where it was continuously dark for long stretches at a time, i.e. at very northern latitudes. Also, they propose that by absorbing more light, blue eyes might have been selected for as a means of protection against things like seasonal affective disorder.