Another Way of Thinking About Ancient Populations (Autosomes versus just Y and mtDNA)

I don't think Neandertals died out at all.
 
No more so than any population that existed from 600,000 to 25,000 years ago.
 
If you tested ANY species of Homo that old, of course they wouldn't match us exactly. The genus has evolved.

Neandertal population size was tiny. Imagine that there were 40,000 of them in Europe. (That number is actually large.  Many aDNA experts believe that there were never more than 10,000 Neanderthals alive at any one time!)

Now imagine that 1 million modern humans come into Europe, during various phases.
 
You mix them together, and you get the 4% of Neandertal autosomes in our populations.

You also get drastically smaller odds that their sex-linked DNA survives over time.

Never, ever forget population size.

This "study" has been replicated in modern times.
 
Imagine 100 men are marooned on an island. 4 have the surname "Rarityrareness."

After generations, it is likely that the sons of those 4 will have a generation or two that produces only daughters. In fact, it's almost certain.

So the odds are that there will be no Y chromosomes of the Rarityrareness males.

But did they survive? Yes. Their descendants through their daughters are very much alive in the population.  And like Neandertals, large percentages of their genome would survive autosomally, perhaps as high as 80%!

Never ever forget initial and comparable population size. It explains just about every ratio of the newer versus older Y Chromosomes in Europe. 

It explains the lack of Neandertal sex-linked DNA, and it explains the smaller number of the Old Europe haplogroups from small hunter-gatherer populations.