Here is a scholarly treatment of the subject at Nature, and an alternative link to the whole article The timing and spatiotemporal patterning of Neanderthal disappearance for those of us without a subscription.
The NY Times headline is eye-catching, but the whole story is a little more complex. Our knowledge here is still developing with recent discoveries and re-evaluations of previous artifacts, so it may be some time before we can reconstruct the “full story” of human/Neanderthal development and interaction. With all of those caveats, here is my “short version”:
Patrick is correct that for 100,000 years after the appearance of anatomically modern humans (AMHs) around 200 kya, we were genetically, socially, and technologically pretty similar to the Neanderthals, as well as other hominids. But around the time of the Neanderthals’ disappearance, anatomically modern humans had become behaviorally modern humans. Since behavior isn’t fossilized, archaeologists identify behaviorally modern humans through artifacts that indicate symbolic behavior – such as personal decoration (ochre, beads, jewelry), ceremonial burial, figurative art, etc. The first evidence of this is found in S. Africa around 77 kya. More appears in the Levant (Near East) about 50 kya as AMHs migrated out of Africa, and shortly afterward it is found everywhere that homo sapiens traveled. Looking at this pattern, Ian Tattersall goes so far as to say that humanity had two origins: one anatomical, and the other cognitive.
How do the Neanderthals fit this picture? Without question, the previous assumption that humans pushed the intellectually inferior Neanderthals to extinction is incorrect. They actually had a larger cranial capacity than modern humans (1600 cc vs. 1400 cc), so it seems reasonable to guess that they had an overall intelligence close to ours. Some previous scholars argued that Neanderthals did not have the necessary physical vocal apparatus to produce speech sounds like ours, but recent evidence seriously calls that into question. On the language front, a few symbolic artifacts have been attributed to Neanderthals, but these appear very late in their history, during the few thousand years that Neanderthals and H. sapiens coexisted. As the Nature article put it:
“(Neanderthals) did not survive after 41,000–39,000 calBP (Fig. 2b). By comparing the final Neanderthal PDF with those obtained for the start of the Uluzzian at the Cavallo site 23, we can quantify the temporal overlap between Neanderthals and the earliest western European AMHs (Fig. 2b). The difference is significant and ranges from 2,600 to 5,400 years at 95.4% probability. Coexistence has been linked previously with the possibility of cultural transmission from AMHs to Neanderthals, termed ‘acculturation’24, as a means of accounting for late Neanderthal technical and behavioural development. The early presence of AMHs in Mediterranean Europe by 45,000–43,000 calBP (ref. 23) and the potential overlapping time may have acted as a stimulus for putative Neanderthal innovative and symbolic behaviour in the millennia before their disappearance.”
This next bit is my own historical reconstruction, so take it for what you will. By the time that H. sapiens encountered Neanderthals, the latter were already few in numbers and confined to a small pockets of Europe. Genetically, Neanderthals were already doomed by a shrinking gene pool. Although they may have had overall intelligence similar to our own, they were not as far along in language development and still probably used a form of “proto-language” to communicate. It seems likely that their late technological advances were due to contact with behaviorally modern humans, but the fact that Neanderthals interbred with humans (whatever the circumstances) and could copy our more advanced technology speaks to their intelligence and similarity to us. As it stands, though, I don’t think the evidence justifies saying that “Neanderthals were people, too.”
If they had survived, would the Neanderthals have been able to emulate our language skills and become just another strain of modern human? Since they did not survive, it’s impossible to say. It’s kind of like asking what the Cowboys would have done this year if Romo hadn’t been injured.