With waxing, after thoroughly washing the vehicle and drying, a coat of wax is applied and rubbed through the painted surface which penetrates deep into the pores (not visible to naked eye) thus coating the surface which not only shines but also protects paint against weather and dust in air when you drive the car. Cutting short, the painted surface gets a protective coating against weather and abrasive material in air while driving. Waxing is required to be done at least once a month ( how frequent you drive).
Polishing is a process where an abrasive material ( fine abrasive paste) is rubbed over a painted surface which has lost the aesthetic look and shine over time. With polishing a minor amount of painted layer is cut (reducing the thickness of the original painted surface) to get the shine.
But eventually the thickness of the top layer of paint ( measured in microns) is reduced.