There is an excellent documentary called "Decoding the Neanderthal" on Netflix which I highly recommend for anyone interested in science and our history. The topic of interbreeding between Neanderthals and Homosapians is covered in the documentary and offers this very fact as a partial explanation for the extinction of the Neanderthal. Apparently it's quite possible that they were simply "absorbed" into the early Homosapian settlers of Europe. According to DNA trace testing (which is largely what the documentary is about), most modern humans have anywhere from 2% - 10% Neanderthal DNA, with those of European ancestry having the largest proportion of Neanderthal DNA. They also discovered that Neanderthals were more advanced than originally supposed. Interesting stuff.
Often times my sheer idiocy has caused me to joke that I may just be partially Neanderthal. Now I know in all likelihood it isn't just a joke but I may well have the world's highest proportion of their DNA and my wife tends to agree ;)