Modern human tools are more varied and show more innovation through time, and this might have led to lower infant mortality. If the moderns were better able to feed or protect their growing children, then their populations could have expanded faster than those of the Neanderthals. The archaeological record shows that the Moderns expanded through Europe from the South and East, while the Neanderthals moved West, into France, Spain and Portugal. The latest Neanderthal sites are found on the West coast of Portugal, and on the rock of Gibraltar. Poignantly, their backs were against the sea and they had nowhere to go.
Some scientists doubt they ever really went.
They argue that the Neanderthals did not go extinct, but interbred with the interlopers, and may well have contributed genes to later generations. Svante Paabo's team of geneticists in Leipzig managed to sample a tiny fragment of Neanderthal DNA, and showed that it was very different to that of living people. That might be the nail in the Neanderthal coffin, evidence that they were not our ancestors. However, that is just one piece of a huge genetic blueprint. We might yet find a Neanderthal gene that is still around today.