I think this can be applied to much the same domains as simplex or similar algorithms. So optimization problems/OR.
Simplex is provably within the polynomial bounds of complexity, ie: simplex is a so called "easy" problem in complexity theory.
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You're right that simplex is applied to optimization problems. Constraint programming could be seen as a "more generic" optimizer, in that it can handle integer-optimization (simplex cannot handle integer optimization at all).
Of course, integer-optimization is NP complete, which means that constraint programming is innately "inefficient" or "hard" in terms of complexity.
Constraint programming is very broad — it encompasses many types of problems. Some of them are in P, some (most?) are in NP. In principle you can use the simplex algorithm for solving certain constraint programming problems, but mostly when people talk about constraint programming they refer to different techniques.