And yet skinny little men with big brains made more babies and survived more successfully, and here we are.
Cro Magnons didn't beat the Neanderthals by outbreeding them, although that didn't hurt.
The Neanderthal was a cannibalistic creature which nonetheless buried his own dead; that means that a typical Neanderthal saw the living world as neatly divided into two parts i.e. his own family group, and meat. That also means that Cro Magnons never had to face more than 20 - 30 of them at any one time, that was the most major factor. Other factors were fire (a creature with a 6" fur coat was fairly easy to simply light up), dogs (to prevent incursions and defeat most of the Neanderthals' advantage from night vision), and casting weaponry, particularly the atlatl.