“Too busy for” and “too busy with” are two phrases that are commonly used in spoken English. However, the meanings are very different.
If someone asks you, “Want to go watch a movie?”
…and you respond with “Sorry, I can’t. I’m too busy with work.”
Saying you are “busy with” means that something (i.e. work) is taking up all your time and preventing you from doing something else, like watching a movie. The words that come after “busy with” explain what is keeping you busy. Maybe you really want to watch a movie, but you cannot because work is taking up your time.
But if someone asks you, “Want to go watch a movie?”
…and you respond with “Sorry, I’m too busy for movies.”
Saying you are “busy for” means that whatever is keeping you busy is more important than other things (i.e. watching a movie). Compared to “busy with,” saying that you are “busy for” usually gives the feeling that you like being busy and don’t want to do something else. For example, you DON’T want to watch a movie, but want to keep working.