Every topological 3-manifold carries a smooth structure, unique up to diffeomorphism. In addition, "up to homotopy", the maps between them are the same: every continuous map is homotopic to a smooth map, and any homotopy between smooth maps is homotopic to a smooth homotopy (true for any pair of closed smooth manifolds. Probably the inclusion of mapping spaces $\text{Smooth}(M,N) \hookrightarrow \text{Cont}(M,N)$ is a homotopy equivalence, given the right topologies; all one really needs is to be able to homotope continuous maps that are smooth on the boundary to smooth maps, even if the domain is a manifold with corners.)
For closed orientable surfaces of genus $g>1$, $\text{Diffeo}(\Sigma) \hookrightarrow \text{Homeo}(\Sigma)$ is a homotopy equivalence; one can essentially see this by showing that each is homotopy equivalent to the discrete group $\text{Out}(\pi_1)$. (One can check by hand that it's a homotopy equivalence for $\Sigma = S^2$ or $T^2$; but no longer to a discrete group.)
What can we say for the same situation in closed 3-manifolds? Is the inclusion $\text{Diffeo}(M) \hookrightarrow \text{Homeo}(M)$ a homotopy equivalence? Does it at least induce a bijection on $\pi_0$ (i.e., induces an isomorphism on the smooth and continuous mapping class groups?) If not, can we restrict to some subclass (hyperbolic?) for which either of these is true?
(By the computation of mapping class groups of tori, or noting the existence of exotic spheres, this fails miserably for $n \geq 5$. Presumably it also fails miserably for $n=4$.)
The key reference here is Jean Cerf, "Groupes d’automorphismes et groupes de difféomorphismes des variétés compactes de dimension 3", Bull. Soc. Math. France (1959). The full text is available here.
Let $M$ be a closed 3-manifold, $G$ its group of self-homeomorphisms, $H$ its group of self-diffeomorphisms. (All orientation preserving for convenience.) He proves that $\pi_n(G,H) = 0$ for all $n \geq 0$, thus that the inclusion $H \hookrightarrow G$ is a weak homotopy equivalence - assuming Smale's conjecture (now theorem, due to Hatcher) that the inclusion $SO(4) \hookrightarrow \text{Diff}^+(S^3)$ is a homotopy equivalence. (In fact, he shows that the inclusion is a homotopy equivalents, but the arguments are a bit more delicate.)
For a manifold $M$ with boundary, $H$ and $G$ as above should be the automorphisms restricting to the identity on the boundary. The idea is to show that, if you have a decomposition $M = M_1 \cup M_2$, with $M_1 \cap M_2$ a properly embedded surface, having $H_i$ be $(n-1)$-connected in $G_i$ (for both $i$) is equivalent to having $H$ be $(n-1)$-connected in $G$.
Now we want to induct. Say that a manifold homeomorphic to $D^3$ is order 0; and a manifold that decomposes into order $(n-1)$-pieces is order $n$ if we can't decompose it into pieces smaller than order $(n-1)$. But by Smale's theorem, it's true that $\pi_i(G,H) = 0$ for all $i \geq 0$ when $M = D^3$. Because every 3-manifold has finite order (look at Heegaard decompositions), this proves the theorem.
As for why the third paragraph is true, this is Cerf's "Lemma 0", at the very end of his paper.