Dear Victoria: here is a summary of the main comparison results I know of between Grothendieck cohomology (which is usually just called cohomology and written $\newcommand{\F}{\mathcal F}H^i(X,\F)$ ) and other cohomologies.
1) If $X$ is locally contractible then the cohomology of a constant sheaf coincides with singular cohomology. [This is Eric's answer, but there is no need for his hypothesis that open subsets be acyclic]
2) Cartan's theorem: Given a topological space $X$ and a sheaf $\F$, assume there exists a basis of open sets $\mathcal{U}$, stable under finite intersections, such that the CECH cohomology groups for the sheaf $\F$ are trivial (in positive dimension) for every open $U$ in the basis: $H^i(U,\F)=0$
Then the Cech cohomology of $\F$ on $X$ coincides with (Grothendieck) cohomology
3) Leray's Theorem: Given a topological space $X$ and a sheaf $\F$, assume that for some covering $(U_i)$ of $X$ we know that the (Grothendieck!) cohomology in positive dimensions of $\F$ vanishes on every finite intersection of the $U_i$'s.
Then the cohomology of $\F$ is already calculated by the Cech cohomology OF THE COVERING $(U_i)$: no need to pass to the inductive limit on all covers.
This contains Dinakar's favourite example of a quasi-coherent sheaf on a separated scheme covered by affines.
4) If $X$ is paracompact and Hausdorff, Cech cohomology coincides with Grothendieck cohomology for ALL SHEAVES
If you think this is too nice to be true, you can check Théorème 5.10.1 in Godement's book cited below
[So Eric's remark that no matter how nice the space is, Cech cohomology would probably not coincide with derived functor cohomology for arbitrary sheaves turns out to be too pessimistic]
5) Cohomology can be calculated by taking sections of any acyclic resolution of the studied sheaf: you don't need to take an injective resolution. This contains De Rham's theorem that singular cohomology can be calculated with differential forms on manifolds.
6) If you study sheaves of non-abelian groups, Cech cohomology is convenient: for example vector bundles on $X$ ( a topological space or manifold or scheme or...) are parametrized by $H^1(X, GL_r)$. I don't know if there is a description of sheaf cohomology for non-abelian sheaves in the derived functor style.
Good references are
a) A classic: Godement, Théorie des faisceaux (in French, alas)
b) S.Ramanan, Global Calculus,AMS graduate Studies in Mahematics, volume 65.
(An amazingly lucid book, in the best Indian tradition.)
c) Torsten Wedhorn's quite detailed on-line notes, which prove 1) above (Theorem 9.16, p.92) and much, much more.
By the way, @Wedhorn is one of the two authors of a great book on algebraic geometry.
d) Ciboratu, Proposition 2.1 and Voisin's Hodge Theory and Complex Algebraic Geometry I, Theorem 4.47, page 109 , which both also prove 1) above.
Q1: A very simple example is given in Grothendieck's Tohoku paper "Sur quelques points d'algebre homologiquie", sec. 3.8. Edit: The space is the plane, and the sheaf is constructed by using a union of two irreducible curves intersecting at two points.
Q2: Cech cohomology and derived functor cohomology coincide on a Hausdorff paracompact space (the proof is given in Godement's "Topologie algébrique et théorie des faisceaux"). I don't know of an example on a non paracompact space where they differ.
Best Answer
Both cases ($F$ is a sheaf of abelian topological groups or abelian Lie groups) can be treated using the same machinery.
The Yoneda embedding embeds abelian Lie groups as a fully faithful subcategory of the category of sheaves of abelian groups on the site of smooth manifolds, and the embedding functor preserves small limits. I refer to the latter category as the category of abelian smooth groups. This category is complete and cocomplete. It is, in fact, better behaved than the category of abelian topological groups, since the latter category is not an abelian category: a morphism of abelian topological groups can have a trivial kernel and cokernel, without being an isomorphism. On the other hand, the category of smooth groups is abelian.
From now on, we work either with presheaves of chain complexes of abelian topological groups or presheaves of chain complexes of smooth groups.
The category of such chain complexes can be equipped with the projective model structure transferred from simplicial topological spaces respectively simplicial sheaves on the site of smooth manifolds.
The category of presheaves of such chain complexes can itself be equipped with the projective model structure, which can then be further localized with respect to Čech nerves of open covers. Fibrant objects in the resulting model category are presheaves of chain complexes that satisfy the homotopy descent condition.
The fibrant replacement functor computes the (hyper)cohomology of sheaves. More precisely, if $F→RF$ is a fibrant replacement of the sheaf $F$, then $H^i(X,F)=H^i(Γ(X,RF))$.
The resulting cohomology theory is essentially (a reformulation of) the Segal–Mitchison cohomology in the topological case, or its smooth version by Brylinski.
In particular, the cohomology group $H^i(X,F)$ is by definition a smooth (respectively topological) group, since $H^i$ is computed in the category of smooth (respectively topological) groups.