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It's more likely that their culture did them in. Neanderthals were slow to adopt new ideas and new technologies. This is evidenced by their stone tools. They stuck to the same tool designs for thousands of years. It wasn't until modern humans arrived that they started to toy with new designs. It's probably a fair guess that they didn't change other aspects of their culture either. This would have posed a problem if the environment changed too rapidly.

Rapid change could have resulted from the activities of the modern humans. In particular, the moderns probably used fire to alter the environment. Many hunter-gatherers burn off undesireable plants to allow more valuable plants to thrive. For example, here in Southern California, the Indians were all hunter-gatherers. They burned off the chapparel to allow sagebrush to grow. They then collected the sage seeds to grind into meal. This destroyed habitat for some animals and created it for others. Any species depending on the previous mix of game animals would suffer as a result of the change. Modern Humans changing the environment through fire in Europe and the Middle East would have affected the Neanderthals, but the Neanderthals were loathe to change their habits to adapt; even though they had the physical and intellectual ability to do so.