I don't know if it's bad form to reply to something this old, but I stumbled on this question because I've wondering about the negative result for a couple of days now. That is, for an explicit example where Cech cohomology differs from (derived functor) sheaf cohomology. In case anyone is also curious about this, I did find an example buried in pages 177-179 of Grothendieck's classic "Tohoku" paper "Sur quelque points..."
Donu Arapura
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