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...". Perhaps I can say a few words about it since it is surprisingly simple. Take X to be the affine plane over a field, and let $Y\subset X$ be the union of two irreducible curves meeting at two distinct points. Let K be the kernel of the restriction map $Z_X\to Z_Y$ of the Z-valued constant sheaves on the Zariski topology. Then he shows that $H^2(X,K)=0$ H^2(X,K)=Z$but that the Cech group$\check{H}^2(X,K)= Z$0$. (I wrote this backwards previously, sorry about that.)
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..."points...". Perhaps I can say a few words about it since it is surprisingly simple. Take X to be the affine plane over a field, and let $Y\subset X$ be the union of two irreducible curves meeting at two distinct points. Let K be the kernel of the restriction map $Z_X\to Z_Y$ of the Z-valued constant sheaves on the Zariski topology. Then he shows that $H^2(X,K)=0$ but that the Cech group $\check{H}^2(X,K)= Z$.