Precipitation is the process of formation of a solid in solution during a chemical reaction. The solid has a higher density than the solvent (water) and thus it settles to the bottom of the container and is called a precipitate.
Crystallisation is a process of forming solid crystals from a saturated solution of the substance. As a solid crystal separates from a solution of the substance, this process is similar to precipitation. However, in crystallisation no chemical reaction takes place.