Isotopy in 3-manifolds Assume $\Sigma_1$ and $\Sigma_2$ are two embedded compact surfaces 
(say orientable) in an orientable 3-manifold $M$. Assume $\Sigma_1$ and $\Sigma_2$ 
are homotopic in $M$. Then are they isotopic? 
 A: If $\Sigma_1 \hookrightarrow M$ is an embedded $\pi_1$-injective surface, then any homotopic embedded surface will be isotopic to $\Sigma$. As Ryan and Allen point out, this is due to Waldhausen for incompressible surfaces of genus $>0$, and to Laudenbach for 2-spheres, together with the Poincare conjecture. 
If $\Sigma_1$ is not incompressible in $M$, then there exists $\Sigma_2$ which is homotopic to $\Sigma_1$ but not isotopic. The point is that one may compress $\Sigma_1$ to get a surface $\Sigma'\hookrightarrow M$ which has smaller genus, and then reembed the 1-handle (in the same homotopy class) in a knotted fashion to get a non-isotopic surface $\Sigma_2$. Misha observed this in the comments on Ryans question for tori in $S^3$, but it holds more generally. 
There's an intermediate case of $\Sigma_1\hookrightarrow M$ which is incompressible and not $\pi_1$-injective. By the loop theorem, this can only occur if $\Sigma_1$ is 1-sided in $M$, which implies that the surface is non-orientable, so does not fall under the purview of your question.  I'm not sure if homotopy implies isotopy in this case - I suspect there are 1-sided Heegaard surfaces which are homotopic but not isotopic, but I don't know examples off the top. 
A: No, generally they're not. 
For example, there's only one homotopy class $S^2 \to \mathbb R^3$ but there's two isotopy classes of embeddings (given via how the embedding orients the compact 3-manifold it bounds). 
edit: I think if your 3-manifold is irreducible and if your maps $S^2 \to M$ are not null homotopic then the answer is likely yes.  But if your 3-manifold is say a connect sum of lens spaces then I suspect it's false but I haven't come up with a nice example yet.  As Allen points out in the comments below, a connect-sum of lens spaces won't work, at least not when your surface is a sphere. 
edit2: As Misha Kapovich points out, for irreducible 3-manifolds and incompressible surfaces homotopy implies isotopy.  This is an old theorem of Waldhausen's.  "On Irreducible 3-manifolds which are sufficiently large" Ann. of Math (2) 87 (1968) 56--88. 
