PDEs on manifold: what changes from Euclidean case?

The Poincaré inequalities for $H_0^1$ and for $H^1/\{\textrm{constants}\}$ will remain, because they depend only on connected open sets having rich families of connecting curves. Second order estimates, such as bounding $\|u\|_{H^2}$ by $\|u\|_{L^2}+\|\Delta u\|_{L^2}$ will be messier, because switching to $\mathbb R^n$ produces undesirable terms (chain rule for second derivative), but I think those will be absorbed. To be safe, you should not take things for granted, but either check that the proof carries out or consult the manifold-oriented literature.

Things do change a lot when one considers maps rather than scalar functions. For example, the Laplace equation for maps between manifolds is nonlinear, and the regularity of its solutions is an entirely different story from what happens in the flat setting.

Perhaps the main difference between PDE on manifolds and PDE of Euclidean domains is the shift of focus: people simply ask different kinds of questions. Parabolic problems are now less about diffusion of matter and more about evolving metrics on the manifold toward some canonical type. Existence or nonexistence of solutions is of more interest if it detects some topological or smooth invariants of the manifold.

(Hopefully, Willie Wong will find time to give a better answer.)

As a first approximation, you are already doing PDEs on a manifold if you are studying variable coefficient PDEs. Once you know what you are doing, analytic arguments on closed manifolds tend to be technically simpler and cleaner because there is no boundary. Obviously, you have to be careful when you want to import global considerations from flat domains to manifolds. But this does not mean that the manifold case is more difficult; boundaries can cause a lot of trouble in $\mathbb{R}^n$.