Just because we carry the same Foxp2 gene, it doesn't mean that Neandarthals directly evolved into Humans.
For one, there is evidence of Neanderthals and Homo Sapiens Sapiens living together, and perhaps even mating together. If HN. evolved into HSS., then neither could be extant alongside the other. Populations evolve, not individuals, and if HN. evolved directly into modern humans, then by the time we came to be, HN. would not exist as a species.
The only explanation, and the most logical one at that, is that the Foxp2 is a shared derived characteristic of both species. Both species evolved from a single common ancestor which also had the same gene.