Adding to @David's answer, there has been some recent work looking at Neanderthal DNA in the context of Denisovan and modern human populations.
David Reich's group [Sankararaman et al. Nature 2014] found Neanderthal-derived alleles in genes that affect skin and hair (specifically keratin filaments), as well as some that confer risk of disease. Another cool finding:
... there is an approximately fivefold reduction of Neanderthal ancestry
on the X chromosome, which is known from studies of diverse species to
be especially dense in male hybrid sterility genes. These
results suggest that part of the explanation for genomic regions of
reduced Neanderthal ancestry is Neanderthal alleles that caused
decreased fertility in males when moved to a modern human genetic
A pair of papers from the groups of Joshua Akey [Vernot and Akey. Science 2016] and Tony Capra [Simonti et al. Science 2016] found associations between Neanderthal alleles and various phenotypes including depression, hypercoagulation, and tobacco use. These results are nicely summarized here.