Here, lots of ideas have been put forward — a sure sign that no one knows. Perhaps they died of mad Neanderthal disease, owing to a habit of feasting on one another’s brains. (This has been put forward as a serious hypothesis.) Perhaps they were victims of a changing climate. Perhaps they were “inferior” beings, unable to match our capacity for innovation in the face of adversity. Perhaps their populations became too small, and too sparse, for them to find mates. Or — and this is the most haunting possibility — perhaps they were eventually murdered by their puny cousins. That is, us.To summarize:
- climate change
- competition with humans
- population decline
And I'll note that it's much, much easier to write with the tongue in the cheek than it is to actually speak.
The evidence is, again, that some Neanderthals and humans had sex and fertile offspring, because there are clues from the newly sequences Neanderthal genome that humans carry some of their DNA (actually, only among human lineages that left Africa, as opposed to those who stayed - Yoruba dna Khoi-san, specifically).
So what I propose is that what happened was that when Neanderthal men laid eyes on the women of the newcomers, they found these human ladies so fair compared to their own brutish women that they summarily lost interest in their own kind, and instead focused on mating with human females.
You can just picture it: supermasculine Neanderthal spots sexy human female who couldn't possibly resist his strength (as if his brutish appearance wasn't perfectly enticing in the first place). This would explain both the new evidence for interbreeding and the extinction of the Neanderthals.