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"By comparing this composite Neandertal genome with the complete genomes of five living humans from different parts of the world, the researchers found that both Europeans and Asians share 1% to 4% of their nuclear DNA with Neandertals. But Africans do not.
This suggests that early modern humans interbred with Neandertals after moderns left Africa, but before they spread into Asia and Europe.
The evidence showing interbreeding is "incontrovertible," says paleoanthropologist John Hawks of the University of Wisconsin, Madison, who was not involved in the work. "There's no other way you can explain this."

This suggestion that humans and Neanderthals "share 1% to 4% of their nuclear DNA," is necessarily false, and one has to wonder if it's not just someone's political agenda masquerading as science.

Humans and Neanderthals do not share 1% to 4% of their nuclear DNA. We share about 99.9% with Neanderthals, the same as shared amongst other humans.

The nuclear DNA differences between humans and Neanderthals is not 1% or 4%, it's one-tenth of one percent. In terms of actual numbers, 3 million "base pairs" out of 3 billion total are different. And that's the same number as differences amongst various modern humans.

If there are some identical DNA base pairs shared by Neanderthals, Europeans and Asians, but not Africans, that number must be much smaller than one-tenth of one percent.

None of which is intended to deny that our horny-goat ancestors were just as likely to hanky-pank indiscriminately as we are. It's just that the results were not as significant as those 1% to 4% numbers suggest.

You are confused about what they are comparing, and no wonder as journalists write about science about as well as they write about other things, which is to say, not very well.

A wolf and a coyote may be some 97% or so identical to a wolf in genetic DNA, but that doesn't mean that one cannot find genetic evidence of wolf ancestry within some coyote populations by way of crossbreeding, and be able to estimate that coyote population as having 1 to 4% wolf ancestry.

Similarly, a human and a neanderthal may be some 99% or so identical in genetic DNA, that doesn't mean that one cannot find genetic evidence of neanderthal ancestry within some human populations by way of crossbreeding, and be able to estimate that human population as having 1 to 4% neanderthal ancestry.

Neanderthals are distinctly non human.

The amount of difference you would expect to see in neutral genetic markers between human populations is AT MOST one third of what you would see when comparing a human population and the NON human neanderthal.

The evidence suggests that while the most distantly related humans shared a common ancestor within the last 100,000 years, humans and neanderthals were different populations some 250,000 years ago.

In morphology and DNA, neanderthals are distinctly not human.

They were most certainly not from a “normal” population of European humans that had rickets and were put in the Stone Age equivalent of a leper colony.

The researchers are only reporting on PARTS of the genome where they expected differences. That's how they come up with those percentages.

I think they are presuming the rest of the genome is pretty much identical.

Wait until they get into epigenetics ~ that's where you'll find the Neanderthal materials all over the place. After all, a mere 20,000 years of contact with life in the midst of an Ice Age is really not enough to elicit the sort of cold climate adaptations modern Northern hemispheric humans exhibit. On the other hand, a full 400,000 years of such contact, as Neanderthals, ought to explain it all quite readily.