In modern English, we tend to use 'dream about' for what we experience when we're sleeping ( "I dream about you every night") and 'dream of' for future goals and fantasies ("I dream of owning a luxury sports car one day").

'dream about' and 'dream of' are generally interchanged, but there can be a subtle difference between the two. "dream about" is being more indirect or less specific. You are being a little more vague on the specifics of what you were dreaming about. "dream of" is being a little more direct or specific of what you were dreaming.

In my opinion, there is no difference and either is equally correct. I'm not an english teacher so perhaps a real teacher would answer differently. If you are speaking in the past tense, you could say, "I dreamt of/about..." or "I dreamed of/about...." Either is correct.