Tim (remote central) points me to a study by Banks, d'Errico and colleagues that reviews the hypothesis suggesting that Neanderthals went extinct due to climatic constrain:
William E. Banks et al. Neanderthal Extinction by Competitive Exclusion. PLoS ONE, 2008.
Despite a long history of investigation, considerable debate revolves around whether Neanderthals became extinct because of climate change or competition with anatomically modern humans (AMH).
We apply a new methodology integrating archaeological and chronological data with high-resolution paleoclimatic simulations to define eco-cultural niches associated with Neanderthal and AMH adaptive systems during alternating cold and mild phases of Marine Isotope Stage 3. Our results indicate that Neanderthals and AMH exploited similar niches, and may have continued to do so in the absence of contact.
The southerly contraction of Neanderthal range in southwestern Europe during Greenland Interstadial 8 was not due to climate change or a change in adaptation, but rather concurrent AMH geographic expansion appears to have produced competition that led to Neanderthal extinction.
Their modelling shows that in the warm and wet GI8 climatic period (c. 38-35,000 BP) Neanderthals could exploit nearly all Europe but in fact were restricted to parts of Iberia, as shown in this map:
Red: surely apt, Pink: probably apt, Grey: possibly apt, Dots: actual Neanderthal sites