Archaeological record of Neanderthals was not dissimilar to that of contemporary modern humans
A paper by Paola Villa and Wil Roebroeks in open access PLoS One has reviewed archaeological evidence for the view that the ‘inferiority’ of Neanderthals to modern humans was responsible for their demise.
Villa and Roebroeks considered a number of commonly-proposed hypotheses for the extinction of the Neanderthals. These included the views that Neanderthals lacked the capacity for syntactic language and symbolic behaviour; that they had only a limited capacity for innovation, and their technology was inferior to that of modern humans; and that they were less efficient hunters than modern humans, with a less varied diet.
In every case, Villa and Roebroeks were able to demonstrate that the available archaeological does not support the hypothesis. For example, there is evidence for Neanderthal symbolic behaviour that is as strong as anything claimed for modern humans from the African Middle Stone Age, even if the controversial Châtelperronian data is ignored. While the pace of technological change was slow, the same could be said of the later part of the African Middle Stone Age after modern humans emerged. There is also good evidence that Neanderthals were efficient hunters, and that they enjoyed a varied diet.
The question, then, is if Neanderthals were not technologically and cognitively disadvantaged over modern humans, why did they die out? Villa and Roebroeks suggest multiple factors played a part including low population densities in comparison to the incoming modern humans; and possible male sterility arising from interbreeding with modern humans. Eventually, they suggest, the Neanderthals were swamped and assimilated by the increasing numbers of modern immigrants.
Villa, P. & Roebroeks, W., Neandertal Demise: An Archaeological Analysis of the Modern Human Superiority Complex. PLoS One 9 (4), e96424 (2014).