The idea of Homo Sapiens Sapiens taking out Homo Sapiens Neanderthalensis is ridiculous, the average Neandertal could have taken the average Sapiens and twisted him into a pretzel. With greater brain capacity and much greater strength, Neandertal had a huge advantage, thinking otherwise is simply pure sapiens arrogance. Without the effect of the Pleistocene extinction event, I would have predicted the opposite out come, which means we may owe our very existence to this past event.
It is my understanding that Homo (Sapiens?) Neanderthalis has three major disadvantages when compared to Homo Sapiens Sapiens. We can run better. We have arms better suited to throwing. In other words we have range and mobility on our side should it come to a fight. We eat a wider variety of food (from analysis of bone data from Neanderthals - they ate a higher proportion of meat, much closer to a pure carnivour diet).
Secondly, larger brain capacity does not directly imply higher intelligence. Without external evidence I see little reason to believe Neanderthals were actually smarter than humans.