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Your latter question the answer is generally no. A diffeomorphism $f : S^n \to S^n$ has homotopy class given by its degree $\pm 1$. But the homotopy-through-diffeomorphisms (usually called isotopy) classes are the group of exotic smooth structures on $S^{n+1}$ provided $n \geq 5$.

There are large classes of manifolds for which the homotopy-classes of diffeomorphisms are reasonable. Hyperbolic $n$-manifolds for $n\geq 3$ have the property that homotopy-equivalences are homotopic to isometries. This is "Mostow rigidity". So homotopy-classes of diffeomorphisms are the same things as homotopy-classes of homotopy-equivalences in this case, which is $Out(\pi_1 M)$, since hyperbolic manifolds are $K(\pi,1)$-spaces.

If you generate 3-manifolds via Heegaard splittings there is a sense in which most 3-manifolds are hyperbolic, so the above gives you an answer in one instance of your question.

But in general there's not much known about the forgetful map

$$\pi_0 Diff(M) \to \pi_0 HomEq(M)$$

Perhaps the largest obstruction to understanding this map is that we know so little about $\pi_0 Diff(M)$.

In high dimensions surgery theory gives you some tools.

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Your latter question the answer is generally no. A diffeomorphism $f : S^n \to S^n$ has homotopy class given by its degree $\pm 1$. But the homotopy-through-diffeomorphisms (usually called isotopy) classes are the group of exotic smooth structures on $S^{n+1}$ provided $n \geq 5$.

There are large classes of manifolds for which the homotopy-classes of diffeomorphisms are reasonable. Hyperbolic $n$-manifolds for $n\geq 3$ have the property that homotopy-equivalences are homotopic to isometries. This is "Mostow rigidity". So homotopy-classes of diffeomorphisms are the same things as homotopy-classes of homotopy-equivalences in this case, which is $Out(\pi_1 M)$, since hyperbolic manifolds are $K(\pi,1)$-spaces.

If you generate 3-manifolds via Heegaard splittings there is a sense in which most 3-manifolds are hyperbolic, so the above gives you an answer in one instance of your question.

But in general there's not much known about the forgetful map

$$\pi_0 Diff(M) \to \pi_0 HomEq(M)$$

In high dimensions surgery theory gives you some tools.