I am not sure I understand this. I thought the Neanderthal was adapted to the Ice Age, and lost out to Cro-Magnon when the Ice Age receded and Cro-Magnon moved up from Africa.
Half the picture. The Pleistocene is the period of time that encompasses the rise and spread of modern homo. It is one characterised by repeated glacial 'events' where around 30% of the landmass of the world was covered in ice. The fall-out of this is two-fold:
1. The ice was created and held by a severe reduction in rainfall - water was effectively held in 'ice-storage'
2. The areas not in the cold-freeze were opened up in the tropics - perfect for the variety of human we are, but deadly for H. Neanderthalis.
Our species is a creature of the hot areas, the first 'beach-combers' landed in Australia way before the colonisation of Asia or the Americas. Europe was the last continent to be colonised by modern homo, even though it is one o f the closest to Africa. We wandered out of Africa with a predisposition to hot climates and the Ice-Ages opened up more and more of those tropical climes as the water receded and covered the areas that Neanderthals preferred.
This would effectively split the Alps into two zones with modern Northern Italy as the divide. As I pointed out - to move away from the 'homelands' to the East would involve competing with other humans. The move to the West (Gibralter) buys time, but leaves them so isolated that they are locked into a dead end.