Most MIP solvers contain performance heuristics tailored to MIP metaphors. If you're using translation, some CPs could translate into MIP formulations that MIP

Most MIP solvers contain performance heuristics tailored to MIP metaphors. If you're using translation, some CPs could translate into MIP formulations that MIP-solvers happen to not handle that well (these are theoretical worst-case exponential problems, and you can't know a priori if your solver is going to hit the worst case; MIP solvers are fast in general but there's a measure of luck involved in the solution process -- anyone who does discrete optimization for a living knows this). You're also at the mercy of the translation algorithm, which may not possess a library of the most efficient MIP formulations. Furthermore, MIP modeling is an art which takes years to hone (less so today than when I started -- algorithms are getting more and more sophisticated -- but nevertheless, it takes human reasoning and intuition to produce a good formulation). It's always useful have a range of solvers at your disposal and not limit yourself. CP solvers are specialized to CP problems and for simple problems they may work better due to less translation overhead.