[SampleMan]

The objection can always be raised that this is all conjecture or 'just so' argumentation, but here's a fact: We're here; Neanderthals aren't. Obviously we were better at whatever it is we needed to do to make that so.

Or it was just bad luck for them. Yet, the real problem isn't explaining how we might have won out as the prominent species, its explaining how they could have gone extinct globally, in such a short time. Remember, modern humans were hardly filling up the land masses, and mainly abandoned harsh climates, like the ones that Neanderthals survived in for a few hundred thousand years. So where did they go. Even intentional or untentional genocide can't explain full extinction, when modern humans occupied such a small percentage of the land mass.

[AntiGuv]

the real problem ..., its explaining how they could have gone extinct globally

Neanderthals were never global. Their range was limited to Europe, which right there is a sign of their cognitive limitations by comparison to modern humans.

[Alas Babylon!]

One possibility is disease. It is now extimated that up to 90% of American Indians died within 50 years of Columbus's initial discovery (Timeline of European Disease Epidemics Among American Indians). Also, consider how many conatgious diseases spread from Africa. Africa breeds an exceptionally large amount of human disease as it is the birth place of the human species; diseases evolved along with humans for millions of years there.

I think the wave of human settlers appearing in the Middle East and Europe before the Ice Age Maximum of about 35,000 years ago brought hunting pressure and perhaps warfare, as stated by the author, but disease, if carried over to a previously isolated population of humans, like it did to the American Indian, could have easily wiped out our Neanderthal cousins.