Here's a short video recreation of what the first interaction of modern humans and Neandertals may have been like. We know eventually the interaction was fatal for the Neandertals, but they did co-exist in Europe with modern humans for 14,000 years. There had to have been extensive interaction, including mating, trading, sharing goods, or fighting. Scientists have shown that the Neandertals' DNA was significantly varied genetically from Cro-Magons. Perhaps they were a sub-species, and therefore offspring weren't likely viable. Even if offspring did survive, they would have been few, and their DNA would eventually be barely detectable, due to modern humans' more viable and populous genes.
Did Man Kill the Neanderthals?