Cuz is a very informal short form of because. You see it in text messages, songs, and chat: cuz, 'cause, or cos. It means the same as because; it is only shorter and much less careful. Teachers, exams, and job letters want the full word. Who uses it? Friends, young people, casual writing. You would not put it in a formal email. Though actually, cuz is also slang for cousin (“Hey, cuz”). That is a different meaning that happens to look the same. Here we are talking only about the because use.