The current Nature is carrying an article by geochronologist Chronis Tzedakis and colleagues, claiming that Neandertals didn't succumb to climate change. I think the paper's conclusions could be more strongly stated, but it is right on.
Basically, they found a calibration between the radiocarbon dates for late Neandertals and the marine sediment climate record, by the simple expedient of using the radiocarbon dates within the record, instead of the usual calendar calibration. So they can show that none of the proposed last occurrences of Neandertals correspond to any exceptional periods of climate change. Only if Neandertals lived as recently as 24,000 years ago is there any chance of a climatic impact on them, as by that time Europe was starting to descend toward the harsh conditions of the Last Glacial Maximum. But anytime before 28,000 years, and the Neandertals would have been facing a pattern of climate cyclicity very much like they had survived for 100,000 years previously.
Science journalism site Scitizen has an audio interview with study coauthor Katarina Harvati, if you're interested in more. I think the conclusion is pretty simple.
In my opinion, the idea that "climate killed the Neandertals" has always been misleading at best. Neandertals did not suddenly disappear.
Now, can we arrive at a more detailed understanding of the demographic dynamics of late Neandertals? Right now, it is fair to say there is a lot of doubt about when the last Neandertals lived. I've covered the problem with several posts in the last few years, including the recent (< 30,000 BP) date for Mousterian hearths at Gorham's Cave, the redating of Vindija G1 up to 32,000 BP, Paul Mellars' attempts to compress the chronology, the reasons why those attempts were wrong, and probably a few others.
The same might be said of the earliest non-Neandertal Europeans. At the moment the earliest diagnostic non-Neandertal Europeans are from Pestera çu Oase, Romania, dating to around 35,000 BP. That makes a minimum of 3000 radiocarbon years for the "transition," whatever form it may have taken. If we count early Aurignacian sites without diagnostic fossils, we might extend the first non-Neandertals as early as 43,000 BP, giving us a transition of around 11,000 years.
You can see from the beginning that climate was not a credible explanation for any of this. There is no climatic explanation that could cover the required time period, because the climate fluctuated on a much finer temporal scale. There was certainly no climatic shift before the onset of the LGM that could have wiped out human populations, especially considering the potential mobility of Neandertals as they moved into southern Europe -- as they may have done many times before. A "cold snap" that just finished off the last Neandertals in Spain was always a non sequitur: why were they only in Spain in the first place? Why did the "cold snap" have little if any effect on the non-cold-adapted modern humans? Why does nobody ever seem to notice the highly significant climate-forcing events that don't match any "critical events" in human evolution. I'm still waiting for somebody to find the catastrophic effects on human evolution for the last Yellowstone event.
It was not cold that imposed twice or more the mortality rates on young adult Neandertals compared to Upper Paleolithic Europeans.
Tzedakis PC, Hughen KA, Cacho I, Harvati K. 2007. Placing late Neanderthals in a climatic context. Nature 449:206-208. doi:10.1038/nature06117