Is there a simple example of a topological space $X$ with a sheaf $\mathcal F$ such that Čech and derived functor cohomologies don't agree? (I don't have any knowledge of schemes, I want $X$ to be a topological space)
In the paper Pathologies in cohomology of non-paracompact Hausdorff spaces, Stefan Schröer constructs a Hausdorff space which is not paracompact, and for which sheaf cohomology with values in the sheaf of germs of $S^1$-valued functions does not agree with the Čech cohomology (for example - the same is true for other sheaves).
The space is constructed by taking the countably infinite join of disks $D^2$, with the CW-structure consisting of two 0-cells, two 1-cells and a single 2-cell, using one of the 0-cells as a basepoint. Then he takes a coarser topology, whereby the open sets are open sets from the CW-topology, but only those that either don't contain the basepoint, or contain all but finitely many of the closed disks. With this topology the space is not paracompact, but is $\sigma$-compact, Lindelöf, metacompact.... and contractible! Sheaf cohomology is non-trivial however.
An interesting point to note is that this is not a k-space, and the k-ification of this space is the original CW-complex.