Here is an article that could very well be included in John Hawks' impressive collection of Neandertal anti-defamation files.
I am personally rather disinclined to accept the theory of Neandertal inferiority (or its correlate, that modern humans must have had some genetic adaptation that made them superior and facilitated their success). The main reason for leaning in this direction is that history abounds in examples of vanquished and marginalized peoples and it's hard to argue that this was due to any superiority of the people that replaced them. In ancient times all it took was a bad commander in war or getting a vital piece of military tech too late and a whole nation might be destroyed. Or, more simply one might find themselves on the wrong side of the numbers game and big groups of people tend to replace smaller ones even if individual members of big and small groups are not particulary different in any measurable way.
PLoS ONE 9(4): e96424. doi:10.1371/journal.pone.0096424
Neandertal Demise: An Archaeological Analysis of the Modern Human Superiority Complex
Paola Villa, Wil Roebroeks
Neandertals are the best-studied of all extinct hominins, with a rich fossil record sampling hundreds of individuals, roughly dating from between 350,000 and 40,000 years ago. Their distinct fossil remains have been retrieved from Portugal in the west to the Altai area in central Asia in the east and from below the waters of the North Sea in the north to a series of caves in Israel in the south. Having thrived in Eurasia for more than 300,000 years, Neandertals vanished from the record around 40,000 years ago, when modern humans entered Europe. Modern humans are usually seen as superior in a wide range of domains, including weaponry and subsistence strategies, which would have led to the demise of Neandertals. This systematic review of the archaeological records of Neandertals and their modern human contemporaries finds no support for such interpretations, as the Neandertal archaeological record is not different enough to explain the demise in terms of inferiority in archaeologically visible domains. Instead, current genetic data suggest that complex processes of interbreeding and assimilation may have been responsible for the disappearance of the specific Neandertal morphology from the fossil record.