There is a bit more to the story. There was interbreeding, which we can make up explanations for but none have any evidence to support them. Neanderthals were also under a lot of environmental stress. They were very inbred well before Homo Sapiens shared their range. We have no evidence they hunted from range (no arrow heads, for example, and their shoulder joints are not adapted to throwing) but relied on close range attacks. This would have become increasing more difficult in the Ice Age, due to the nature of the terrain and the increasing size of the megafauna.

There is a good case to be made that Homo Sapiens were just lucky to be based in the milder climate of Africa, and by chance being more adapted to hunting megafauna due to our body structure. Homo Sapiens are also very inbred. IIRC there are two main genetic bottlenecks that have been detected in our DNA. We almost didn't make it.