What an interesting article! Of course, there are many possible implications for these results.
Most obviously, Neanderthals and humans would have to have mated, which suggests a few possibilities. It might mean that Neanderthals were much more like humans than has been thought previously. It could also mean that there were pockets of Neanderthals into which only a few humans drifted, leaving little in the way of mating choices. I am not certain of the chronological overlap of the two species, but it would seem that Neanderthals did not die out quickly after humans evolved. And in fact, there is a level on which it would appear that they have not died out at all!
The non-African part of this suggests to me some geographical implications, too. It is possible that there were few or no Neanderthals in southern Africa or Australia. That would explain the lack of their DNA. This might be because they lacked the intelligence to migrate to those places or because they were not well-adapted to those places and died too quickly. Still another speculation is that Neanderthals did not arise at all in the area we consider to be the cradle of mankind, which would explain at least the lack of their DNA in southern Africans in particular. I am not knowledgeable enough about to fossil record to know where their fossils have been found, but that record might support this speculation.