You can use either one to mean you wish/hope for something to happen in the future. "He dreams of / about becoming a famous singer." "We dream of/about better days ahead." If you mean the mental activity that happens when you are sleeping, ABOUT is the better choice. "I dreamed about my grandmother last night." ("dreamt" is also correct.)

First difference is grammatical: Dream of + N Dream of + V-ing Dream about + N Dream about + V-ing IS WRONG Second difference: "about" is について and "of" is のことを. This {very small} difference is the same in think of vs. think about. Some people wouldn't consider it a big deal though.