Kolbert mentions this disturbing but plausible theory for their disappearance:
It had been comforting to think that the Neanderthals were inferior to modern humans—less clever or dexterous or communicative—and that that’s why they’re no longer around. It turns out, though, that the depiction of Neanderthals as hairy, club-wielding brutes—popular ever since the first Neanderthal bones were discovered, in the eighteen-fifties—says more about us than it does about them. With each new discovery, the distance between them and us seems to narrow. Probably they are no longer here precisely because we are. And that only makes the likeness more disquieting.