CP is different in that it is less a sigle method than a framework for individual CS insights into enforcing constraints. This is usually down by a global constraint store and a watchlist of variable for the problem under consideration.

A good example is the famous “ All different” constraint. In MIP you would implement that as a fancy summ, in CP one can use a corollary from Berge’s lemma to get a specialised constraint to just enforce that.

Rule of Thump is that CP excels when the problem is thigh lay constraint and one has many global constraints ( those that apply to all variables).

Another point is that contrary to MIP programs that often are modelled directly using sums and inequalities, CP Programms usually are written in a declarative language such as ZINC, which in my opinion makes it easier to get into the subject.

If you are interested: This series of blog posts builds a CP solver in Julia

https://opensourc.es/blog/constraint-solver-1/

ortoools by Google also contains a CP solver and the doc is quite interesting

https://developers.google.com/optimization/cp

Finally there is the dissertation of Guido Tack, who developed a oss Cp solver as his PhD project ( I think that is the internal solver used by SAP software now)

https://www.gecode.org/papers/Tack_PhD_2009.pdf