When to use SAT vs Constraint Satisfaction?
I think this is a very good question. You could also ask: when to use a SMT solver? I have a feeling it might be hard to determine before modeling the problem and actually running the CSP/SAT/SMT solvers and finding out. It is well known that even different solvers perform very differently on the same instances! My intuition also comes from the fact that there are potentially many ways of modeling a problem. Furthermore, there are many ways of doing search and inference, depending on what type of constraints is used (if the formalism in question allows different types).

Take for example the standard Sudoku puzzle. For the standard $8 \times 8$ puzzle, a modern SAT/CSP solver will solve any such puzzle in an instant. However, I would rather use a CSP solver because it is easier to express my intention. The mentioned puzzle needs $9+9+9=27$ alldiff constraints, and I'm done. There's actually quite a bit of research in the context of CSP solvers trying to solve Sudoku instances of different size. As far as I remember, they don't do any comparisons between SAT, but rather try to determine what kind of inference works the best, or what kind of Sudoku puzzles are hard. I don't know if for say $32 \times 32$ Sudoku puzzles, SAT solvers would be faster than CSP solvers.

Different formalisms are able to capture domain specific information and exploiting it better and in different ways. For a bit more on this, see the answer and the comments here.