I think there were several factors that led to the Neanderthals getting overrun by Homo Sapiens. First - neanderthals, light native Americans, were probably wiped out by disease brought in from Africa and the Middle East that they lacked immunity to. Second - neanderthals inhabited Europe in very low population numbers (due to terrible ice age conditions) compared to that of humans coming out of north Africa in the Middle East and it's a well-known fact that smaller populations invent things less quickly because there's not enough people inventing. Third - archaeological evidence is still pretty fragmentary, for all archaeologist know a warrior tribe of Neanderthals could have conquered a much larger tribe of humans and bred with the surviving women after killing all the men.
One thing that is non-beyond a doubt though is that populations with high percentages of Neanderthal seem to rule the world, coincidence?