OK, so maybe it wasn't volcanoes.
Jean-Pierre Valet and Hélène Valladas in a brand new paper  propose that geomagnetic excursions at around 40,000 and around 32,000 years ago would have weakened the ozone layer, thereby irradiating the Neandertals with extra ultraviolet:
No special attention has been given to the geomagnetic excursions of Laschamp and Mono Lake which are synchroneous with the extinction and were the most dramatic events encountered by the Neanderthals over the past 250 thousand years of their existence. During this period the geomagnetic field strength was considerably reduced and the shielding efficiency of the magnetosphere lowered, leaving energetic particles reach latitudes as low as 30°. The enhanced flux of high-energy protons (linked to solar activity) into the atmosphere yielded significant ozone depletion down to latitudes of 40–45°. A direct consequence was an increase of the UV-B radiations at the surface which might have reached at least 15–20% in Europe with significant impacts on health of human populations. We suggest that these conditions, added to some other factors, contributed to the demise of Neanderthal population.
The paper gives all the information anyone could want about the geochronology of these events, evidenced worldwide in lava flows.
By the way, I was wondering during the volcano post why it is "lava flow" but "ice floe". I guess it's because an ice floe floats. When the ice isn't floating, as in a glacier, then it flows, too. And I don't think "floe" is a verb. The only remaining mystery: Why does a Google search for "ice floeing" bring me ads for "ice flooring"?
The paper's discussion expresses why the geomagnetic anomalies may be relevant to Neandertal extinction.
The most widespread measure of UV intensity from the public health perspective is the UV index. This is a value on a scale from zero to 11, which is a linear (not logarithmic) function of UV radiation intensity, weighted in a specific way by wavelength. The damaging UV-B waves contribute disproportionately to the UV index value. I mention this because the average UV index is readily obtained for most cities in the U.S. and Europe, and is a standard part of weather forecasts during the summer.
A 20% increase in UV-B radiation sounds dire, but it's roughly what you get by driving from Massachusetts to Virginia. It is true that melanoma rates and other complications from excess UV radiation are higher in populations who have migrated from higher to lower latitudes, increasing their exposure to UV. But these rates do not rise to a level that threatens to drive those populations to extinction.
We do worry about ozone depletion as a risk to threatened endemic populations, as they may be critically affected by new stressors. So the question is, how threatened and endemic were the Neandertals? Would this possible stressor have been enough to have tipped them over the edge to extinction?
I think we can answer these questions pretty easily. The Neandertals were a cosmopolitan population that occupied most of Western Eurasia spread across at least 20 degrees of latitude. The latest Neandertals persisted in the areas of Europe with the highest insolation. UV radiation may have been a stressor but it cannot have been decisive in their decline.
- . The Laschamp-Mono lake geomagnetic events and the extinction of Neanderthal: A causal link or a coincidence?. Quaternary Science Reviews [Internet]. 2010. Available from: http://dx.doi.org/10.1016/j.quascirev.2010.09.010