I think because of the massive burning of wood with the newly discovered invention "fire", the Homo Sapiens created climate change that lead to the extinction of the Neanderthal. Isn't that what they teach in public schools now?
Actually, according to Discovery Channel they were less mobile because of body structure, and apparently didn't have the apparatus to communicate well through the spoken word, both major disadvantages when competing with H.S. They also were not as advanced in the use of tools, demonstrated by the fact they stuck with Macintosh, and didn't switch over to the PC with a Windows operating system. Wait, that was another Discovery Channel show.
Suffice it to say, they were significantly different from H.S. Modern man kills over such minor differences such as skin color. Ancient man probably didn't have as developed of a moral structure as modern man, and probably killed them off to eliminate competitors.