The race aspects of the Neanderthal theory, while a logical consequence of Neanderthals hybridization mostly in Europe and Asia, is regarded as "politically incorrect" by some people, and provoking on others. Still, if this part wasn't true, neither would the theory. Therefore, I've decided to describe separate from the rest.
Race and human diversity
There has been a lot of research on racial differences. Most of this research has become politically incorrect, especially racial differences in behavior and IQ.
Since natural selection has converged on the proportions of Neanderthal vs modern human gene frequencies, we can conclude that the retained Neanderthal ancestry, and modern human ancestry is just as successful when kept at the current proportions. If this was not true, the frequencies would not be in stasis. Thus, the retained Neanderthal ancestry is not superior to the retained modern human ancestry, or the reverse. The retained traits are diversity, not superior / inferior.
This argument can be applied to Aspies / neurodiversity vs neurotypical function as well. Aspies are not superior to neurotypicals, and neurotypicals are not superior to Aspies. Both of these types builds up human diversity, and when kept at suitable levels, provides increased fitness to whole populations.
New research indicates that Eurasians have 1-4% of Neanderthal and Denisovan ancestry. There is also some evidence of introgression in Africa, possibly introgressing 5% ancient genes into modern Africans. Yet, we only expect the archaic Eurasian diversity introgressed from Neanderthal and other archaic groups in Eurasia like Denisovan to make a difference. Modern humans evolved in Africa or south Asia, from archaic ancestors, so we expect modern humans to already have a majority of the diversity in Africa and south Asia. The contribution from Neanderthal and Denisovan must be similar given the very ancient division of Africa from Eurasia of almost 2 million years according to Aspie Quiz data.
The argument about superiority can be applied to races. Individuals, on average, in some race can not be superior to any other race. This is because the only available diversity is the diversity introduced from archaic Eurasians like Neanderthal and Denisovan, and modern humans, and this diversity is identical regardless of race, and thus it follows that individuals in one race cannot be superior to individuals of some other race.
IQ tests are primarily built in order to measure desirable traits in a culture. A couple of decades ago, these desirable traits were mostly related to Neanderthal splinter-skills, and were measured with non-verbal tests like Ravens Matrices. As social traits become more valued, IQ tests today usually measure verbal abilities, and thus are loaded on different Neanderthal traits. The reason why IQ tests are correlated to educational success, and lots of other measures has to do with how human diversity is distributed. Human diversity is not uniform, it is clustered on two distinct factors that originate from Neanderthal and modern humans. All Neanderthal traits are still correlated with each others to varying extents, the most correlated being related to social behavior and communication traits. When IQ tests measure Neanderthal splinter-skills, or some other Neanderthal trait complexes, they automatically become correlated with anything that has to do with human diversity. That's why the context of IQ tests could shift and they still seem to be valid.
This is actually the primary reason for how results of IQ tests differ between races. Races that contain lesser Neanderthal ancestry will have less of the Neanderthal splinter-skills that makes individuals score high on IQ tests. The lowering of the IQ gap between African Americans and European descent can quite likely be explained by how IQ tests have evolved from measuring mostly nonverbal Neanderthal traits to now mostly measuring verbal ability.
How then do we explain the different success of populations as a whole, with populations with Neanderthal heritage being more successful? This is related to why the Neanderthal traits entered the modern human population in the first place, and stayed as balanced selection. It is obvious that natural selection favored a certain level of Neanderthal ancestry because these traits confered an advantage to the population as a whole. It is called hybrid vigor.