At the risk of dragging this topic back to the original question, I have just finished reading a very interesting book on Neanderthals, the conclusion was, as is often the case, we do not know definitively what happened to Neanderthals.
Remember fossil finds of Neanderthals and other pre Homo-sapien humans are incredibly rare, numbering in the dozens. From this small number of fossils and other evidence we have to try to piece together a story of events lasting hundreds of thousands of years. It is inevitable that there will be conflicting interpretations but the authors believed that the Neanderthal population was declining before modern humans evolved and arrived in Europe. They say that rapid climate change destroyed much of the habitat which enabled Neanderthal hunting techniques to be successful. The Neanderthals were stronger than modern humans, and had a brain of similar size, possible even larger, but modern humans evolved to be chase hunters - they could run for hours after prey exhausting the prey and this was best suited to the open savannas where Humans evolved. Neanderthals were stronger and heavier and this was best suited to ambush hunting, requiring forest cover. As the climate changed, forests gave way to open grassland meaning that Neanderthals were becoming more isolated in their own enclaves.
There is also some evidence that Neanderthals lived in smaller kinship groups, rather than the larger tribes of the humans, this resulted in Neanderthals isolated groups being less able to cope with rapid change, they had less critical mass to fall back on. There is some evidence of family interbreeding in later Neanderthals not apparent in earlier groups, implying they were becoming isolated.
It seems that when the two species finally met in Europe Neanderthals were in serious decline anyway and humanity was growing, populations were so sparse that it is hard to see why humans would have needed to murder them, maybe there were local conflicts, there appears to have been some interbreeding, but fundamentally it seems the Neanderthals just got unlucky with environmental changes at the wrong time and humans got lucky. If events had been different it might be Neanderthals contemplating why the smaller and nimbler Sapiens died out and they didn't!
On a personal level I really wish that some isolated groups of Neanderthals had survived, imagine the earth now with two distinct species of human living side by side. We were sop close to each other in appearance and must have had a reasonably recent evolutionary ancestor, this is a fascinating subject and I intend to read more about it.