MIP solvers work themselves through the search tree using all sorts of clever tricks, constantly running LPs on relaxations (approximations) of the problem to prune the tree. They can be very efficient, but the constraints need to be linear for the LPs to work.
CP solvers are less sophisticated in their search (although some are getting more clever [1]), but they instead delegate to graph-theoretical algorithms [2] that can efficiently deal with more high-level constraints, for instance "I only want solutions where all these variables have different values", which in the MIP world would translate to tons of auxiliary linear constraints and variables that would slow down your solver.
So it really depends on how you write your model, and it's not even always obvious whether MIP or CP is better – which is why Minizinc is so useful. There are even attempts to automatically choose the algorithm to use based on the problem structure [3].
CP solvers are less sophisticated in their search (although some are getting more clever [1]), but they instead delegate to graph-theoretical algorithms [2] that can efficiently deal with more high-level constraints, for instance "I only want solutions where all these variables have different values", which in the MIP world would translate to tons of auxiliary linear constraints and variables that would slow down your solver.
So it really depends on how you write your model, and it's not even always obvious whether MIP or CP is better – which is why Minizinc is so useful. There are even attempts to automatically choose the algorithm to use based on the problem structure [3].
[1] https://github.com/chuffed/chuffed [2] http://www.emn.fr/x-info/sdemasse/gccat/sec5.html [3] https://github.com/CP-Unibo/sunny-cp