How equivalent are the homeomorphism and diffeomorphism groups of 3-manifolds?

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.

Warning: This should be a comment on Mike Miller's answer but I'm not allowed to comment.

The paper cited by Miller is a conference proceeding announcing Cerf's result but there are many more details in Chapter III of the published version of Cerf's thesis: Topologie de certains espaces de plongements Bull. Soc. Math. France 89 1961 227–380 (also in French).

It's also worth mentioning a potentially confusing historical point. In his 1959 announcement, Cerf cites a forthcoming paper by Smale proving what we now call Smale's conjecture. It seems that, by 1961, it became clear Smale's proof didn't work and Cerf's results are accordingly downgraded to theorems conditional on Smale's conjecture. Then Hatcher proved Smale's conjecture in his 1983 paper so we can now use Cerf's results about this question.

One last comment: Cerf's results in those papers also include information on the local situation. For instance, for any neighborhood $U$ of the identity in $Homeo(M)$, there is a smaller neighborhood $V$ such that any diffeomorphism in $V$ is smoothly isotopic to the identity through diffeomorphisms staying in $U$. This does not follow from injectivity of $\pi_0(Diff) \to \pi_0(Homeo)$.