Update - 8/24/2011
Upon a request from "Kate," one of our followers, we did a little further research on her comments reference how did the Neanderthals become extinct and came up with three possible explanations.
1. Could Neanderthals and modern humans breed thus ending the distinctive appearance of the Neanderthals? DNA samples do not show that homo sapiens contain any DNA in our biological make-up that would suggest this as an alternative though it must be remembered only five to seven bone fragments from Neanderthals have been found containing enough material to do a DNA match. A very tiny sample for this sort of research since Neanderthals roamed over a period of 200,000 years. Perhaps in the near future there will be enough evidence for serious research into this concept of breeding. Of course, another theory is both species interbred but much like horses and donkeys their offspring would not be fertile and thus not able to reproduce.
2. Much like when Europeans first came to the new world they carried certain diseases the natives had no immunity to and thus wiped out entire villages. Modern humans who actually did co-exist with Neanderthals for roughly 50,000 years may have inadvertently exposed these more primitive (though the Neanderthals lasted longer than present humans have so far) people to a biological nightmare thus wiping out the already dwindling number of Neanderthals.
3. A change of climate may have produce the catalyst needed to allow the Neanderthals to go extinct. A new ice age was taking over Europe and because of the shorter legs and broader chests compared to the more wiry modern humans the Neanderthals could not adapt to the changing weather patterns. With more snow covering their 'hunting' forests the Neanderthals could not compete and possibly their hunting grounds would no longer support them.
Any of the three above concepts could be true and a shame for one of our closest ancestors but one thing we, at J and L, have not found is that modern humans intentionally hunted the Neanderthals to extinction.