It can make integration of certain functions a heck of a lot easier.
For example in rectangular coordinates becomes in polar coordinates.
Things can get very complex if you insist on working in rectangular coordinates in certain processes or contexts.
For example, we use double and triple integration to find the area of a region under a surface, or the volume of a solid in 3D space. If that region is a circle or is composed of circles, or if the solid is a sphere or a hemisphere, then the integration becomes very complex and very nasty in rectangular coordinates, but if you change the problem into the polar domain, things become a lot easier, and the functions are elementary to integrate.
Cylindrical and spherical coordinates are used to exploit a certain geometry. In addition to the examples given above the classical is locating points (or cities) in the world. There are given by longitude and laditude. Why, because the world is essentiall spherical and longitude and laditude are two angles.