[W]hen we look at the complexity of the material record within Africa, I think it is fair to say that Neanderthal behavior fits comfortably within the continuum represented by MSA [Middle Stone Age] people. "Behavioral modernity" is broadly shared, and doesn't clearly track lines of biological differences. Rachel Caspari and Sang-Hee Lee's work on mortality differences are another concrete illustration of the ways that material culture and behavior do not track with anatomy in these populations.
I'm not convinced that this is accurate, even accounting for the possibility that Hawks may be really referring to the Middle Paleolithic (i.e. from 300,000 years ago up to about 30,000 years ago) rather than the Mesolithic a.k.a. Middle Stone Age (starting around 20,000 years ago until about 12,000 years ago), as terminology for pre-historical archeology is anything but settled.
Yes, there are modern humans who may have used similar tools at times and may have engaged in similar activities. But, there seems to be evidence that from the beginning, modern humans were better hunters and had a much greater capacity to adapt their behavior to their circumstances.
Neanderthals hunted in a way that left them as one part of the local ecology. There were still mammoths, whooly rhinoceroes and other large animals in Europe after hundreds of thousands of years of cohabitation with Neanderthals.
In contrast, everywhere outside Africa that they appeared, in Australia, in Eurasia, in the Americas, and on islands, modern humans rapidly (i.e. within a few thousand years on a given continent) extinguished most megafauna. One must conclude that modern humans were far more effective hunters. African megafauna probably survived to a greater extent only because they co-evolved with modern humans and hence learned to develop defenses against them bit by bit as modern humans became more sophisticated.
Modern humans also had a wider diet, which included many plants and small animals, than Neanderthals, who appear to have subsisted primarily on large animals that they hunted. This diet would have made a much greater knowledge of local flora and fauna necessary and would have required a much wider range of hunting and gathering techniques. Hunting a rabbit, gathering shell fish, fishing, hunting large game, collecting and preparing edible insects, identifying and preparing edible barks, collecting berries, collecting wild grains and collecting mushrooms each took different skills and knowledge. Yet, any given population of modern human hunter-gathers did many of these things to feed themselves. We see evidence of this in artifacts like bone harpoon points from Katanda, Zaire, dating to perhaps 80,000 years ago, something Neanderthals never appear to have developed. It also isn't obvious that Neanderthals ever developed shoes or hats.
Perhaps because of these differences, modern human populations expanded to worldwide and local population density levels far in excess of those of Neanderthal, long before they domesticated plants and animals. Neanderthals also apparently didn't manage to leave the Greater European sphere at any point in their hundreds of thousands of years outside Africa, for reasons that aren't entirely clear, while modern humans did so almost immediately upon leaving Africa. Modern humans were present on four more continents than Neanderthals ever were within 40,000 years.
Modern humans in riverine and coastal environments developed particularly sophisticated pre-domestication material cultures (e.g. Greenland Paleo-Eskimos, Jomon culture in Japan, emerging aquaculture in Southeast Australia, the Native Americans of the Northwest). In contrast, there isn't much evidence that there were ever all that many Neanderthals in existence at any one time, and there is no evidence that Neanderthals with access to fish and seafood resources ever exploited those food resources to develop more complex societies and material cultures.
Modern humans also appear to have had a far less static material culture than did the Neanderthals. The evidence that Neanderthal material culture changed over time, or from one sub-environment to another seems weak. They never advanced beyond the Mousterian stone tool kit (although some would argue that the Chatelperronian tool kit, from 35,000-29,000 years ago, does represent some imperfect imitation of modern human technology, even though it was tool little, too late). Modern humans may have used similar tools briefly, perhaps assimilating the technologies used by the Neanderthals, they innovated and the Neanderthals did not. In contrast, Upper Paleolithic archeology shows new waves of technological developments ever ten thousand years or so. While this is hardly a hot bed of innovation, it contrasts markedly with Homo Erectus and the Neaderthals who were static in their material culture over periods of hundreds of thousands of years.
Even after something like a thousand years of cohabitation in Europe at particularly locations with modern humans, there is little evidence that Neanderthals ever adopted significant cultural innovations from modern humans.
This is particularly notable because there are multiple instances of dramatic cultural borrowing in human history. Most former hunter-gatherers of the Kalahari and Australia rapidly became herders. The Plains Indians of North America likewise rapidly adapted to use horses, herd sheep, make wool textiles and use guns. The Cherokee Indians of the Southeastern United States quickly developed their own script for their language based on the ideas behind the scripts that they encountered, and the neighbors of the Sumerians and Egyptians did likewise. The Maoris of New Zealand rapidly adopted firearms. The people of Oceania made the Sweet Potato a staple crop after what appears to be just a single contact with South America that left no genetic trace. The aboriginal Australians adopted the Dingo into their society after another likewise feeble connection with the outside world.
There is even some evidence in observations of higher primates of cultural borrowing.
If Neanderthals were separated from modern humans more by how they lived than by their cognitive abilities, one would have expected Neanderthals to rapidly adopt modern human material culture innovations.
I'm also not convinced that expressive and symbolic material culture in the Neanderthals was rivaled by that of parallel modern humans. The amount and quality of the art of the earliest modern humans exploded onto the scene with the Upper Paleolithic Aurignacian culture when they arrive to replace archiac hominids that come before them. Neanderthals didn't make paintings or sculptures, comparable to that of modern humans. Neanderthal arts, to the extent that there was any, was a pale shadow of what their modern human counterparts produced.
These signs all point to Neanderthals as far more biologically locked into to their culture than modern humans. Perhaps their cultural practices were more hard wired than those of modern humans. Perhaps they just weren't as smart. There is no dishonor in being developmentally disabled when a more advanced brain hasn't evolved yet.