While the rest of the body can be protected with bodily hair, animal pelts, grasses, tree leaves, dirt (or something like ochre mentioned in the article), etc.

While the rest of the body can be protected with bodily hair, animal pelts, grasses, tree leaves, dirt (or something like ochre mentioned in the article), etc. the eyes is the only body part that couldn't be protected from UV back then. When compared to Cro-Magnon, Neanderthals had bigger (and most probably light colored) eyes adapted to the lower light of their more Northern areas. So quick rise in UV (especially if it was not accompanied by the similar scale rise in the total amount of visible light to trigger eyes f-stop equivalent) would have made for noticeable chronic damage to retina and other parts of the eye (googling UV eye damage brings a lot of possibilities), etc. which would be one more factor against them in competition with the Cro-Magnon whose eyes had evolved in the higher UV environment of Africa.