It is similar to the difference between 'think of' (cursory thoughts, toying with ideas) and 'think about'(think deeper into something).
not a teacher
Agreed, both are correct. The expression "dream of" usually refers to wishes and desires while "dream about" describes something you had a dream about.
Wouldn't dream of, is an expression having specific use, it uses
dream of in the sense of "remotely think of doing something even if the opportunity came
For example:
Even if a letter were lying open on my desk, I wouldn't dream of reading another person's letter .
Even if it were a serious mistake, I wouldn't dream of hurting a child.