Modern humans appeared during the Gottwig Interstadial, a brief warm period 45,000 years ago. So when things returned to cold, why didn't they die out? Simple, sapiens adapts primarily with technology, not genes. It is not that sapiens had better technology, it is that sapiens used technology to adapt. So sapiens used the needle and had better fitting skin garments allowing them to survive even though they were less well adapted to that environment. Neanderthal culture was very, very conservative and traditional. A Neanderthal spent his life trying to imitate the idle hand ax or point whereas sapiens spent their lives trying to invent new tools to make their lives easier.
In the same way, during Upper Paleolithic times, humans, both sapiens and Neanderthals, were preserving food better and making survival a bit easier. With these relaxed conditions, this new environment, sapiens' reproductive ability actually meant that sometimes humans could breed and raise children any time of the year. But if you are only fertile once a year, this means nothing. So sapiens took advantage of better technology while Neanderthals could not.
As for your statement about difficulties in child birth, Neanderthal women had a much wider pelvis than sapiens women so there would have been less difficulty. The actual distance between the pubic bones was greater. Additionally, Neanderthals carried a gene making conception easier and this gene was passed on to European women and exists today. So, if I am correct, not only were sapiens women able to conceive more often but when they absorbed this conception gene from Neanderthals, they became better more efficient at it.