CAT is the exam conducted by the IIMs in Nov every year. CET is conducted by the regional colleges eg. Maharashtra, Gujarat, Rajasthan, Tamil Nadu, etc and each have their own CET exams. In mumbai, the mumbai university awards u MMS instead of MBA for many colleges.
This is very commonly asked questions. As already the difference have been explained, I would like to explain you the difference between Cat and CMAT. Just take a look at this picture given below.