To oversimplify, the main reason that Europeans have lower percentages of Neanderthal admixture than Asians is African migration to Europe in the last 20,000 years, and not natural selection against Neanderthal variants or "basal European" admixture from Southwest Europe. This is encouraging as the notion of "basal Europeans" with little or no Neanderthal admixture in Southwest Europe never made much sense to me.
Several studies have suggested that introgressed Neandertal DNA was subjected to negative selection in modern humans due to deleterious alleles that had accumulated in the Neandertals after they split from the modern human lineage. A striking observation in support of this is an apparent monotonic decline in Neandertal ancestry observed in modern humans in Europe over the past 45 thousand years. Here we show that this apparent decline is an artifact caused by gene flow between West Eurasians and Africans, which is not taken into account by statistics previously used to estimate Neandertal ancestry. When applying a more robust statistic that takes advantage of two high-coverage Neandertal genomes, we find no evidence for a change in Neandertal ancestry in Western Europe over the past 45 thousand years. We use whole-genome simulations of selection and introgression to investigate a wide range of model parameters, and find that negative selection is not expected to cause a significant long- term decline in genome-wide Neandertal ancestry. Nevertheless, these models recapitulate previously observed signals of selection against Neandertal alleles, in particular a depletion of Neandertal ancestry in conserved genomic regions that are likely to be of functional importance. Thus, we find that negative selection against Neandertal ancestry has not played as strong a role in recent human evolution as had previously been assumed.Martin Petr, Svante Pääbo, Janet Kelso, Benjamin Vernot, "The limits of long-term selection against Neandertal introgression" bioRxiv (July 4, 2018) doi: https://doi.org/10.1101/362566
UPDATE: A comment suggests that I have misinterpreted the paper somewhat and I am looking into that possibility.