In most cases, we say dream 'about' for a real dream that you have while sleeping, and dream 'of' for a fantasy or ambition. I dreamt about my grandfather last night. I dream of sailing round the world in my own yacht.

The same difference as there is between "about" and "of". You can look this up online. Many people have already written about it.

As far as I know, there are no differences. It's kinda like "I think about you" and "I think of you", both are correct. I, however, prefer "of" just because it sounds more personal/intimate in certain cases, but it might just be me :P