big picture - Idea of presheaf cohomology vs. sheaf cohomology

Let $X$ be a topological space and $U$ an open cover of $X$.

In this thread Angelo explained beautifully how presheaf cohomology (Cech cohomology) relates to sheaf cohomology:

The zeroth Cech cohomology functor $tilde H^0(U,-):Pre(X)to Ab$ from presheaves on $X$ to abelian groups is left exact and its right derived functors coincide with the cohomology $tilde H^n(U,-)$ of the chech complex. So one may interpret Cech cohomology as derived presheaf cohomology.

On the other hand the inclusion $i:Shv(X)to Pre(X)$ of sheaves on $X$ into presheaves is left exact and the diagram
[
begin{array}{rcl}
Shv(X)&xrightarrow{Gamma_X}&Ab\
isearrow&&nearrow tilde H^0(U,-)\
&Pre(X)&
end{array}
]
commutes. Let $Fin Shv(X)$ be a sheaf. The derived functors $H^n(X,F)$ of the left exact functor $Gamma_X$ are called sheaf cohomology. They are in general not equal to the derived functors $tilde H^n(U,-)$ (Cech cohomology). The relation between the two is the spectral sequence

$$
E_2^{p,q}=tilde H^p(U,H^q(-,F))Rightarrow H^{p+q}(X,F)
$$

induced by the Grothendieck spectral sequence.

• What is the general picture behind this?


• In this thread it is explained why the presheaf $H^q(-,F)$ in the spectral sequence sheafifies to zero for $qgeq 1$. How can one interpret this?


• For $X$ locally contractible and $F$ the sheaf of localy constant functions, sheaf cohomology equals singular cohomology and for a cover $U$ with two open sets the spectral sequence is just the Mayer Vietoris sequence. How does this fit into the general picture?