It wasn't Neanderthals (Homo neanderthalensis) that bred with Homo sapiens but a hybrid, Homo sapiens neanderthalensis. Most all of the Neanderthal genetic material was apparently eliminated and quite possibly what distinguishes that species from our own is the capacity for language. 
Lol, maybe Trump has more Neanderthal DNA than most. 
https://geneticliteracyproject.org/2014/02/04/were-neanderthals-a-different-species/