There is no difference between dreamed and dreamt. Both are considered correct, and both function as the past tense and past participle of the verb dream. Dreamed is preferred in all main varieties of English, but dreamt is especially common in British English; while American writers use dreamt about a tenth as often as dreamed, British writers use dreamt about a third of the time.
Dreamt is more often used in the figurative senses of the word—especially in the phrase dreamt up—while dreamed is more likely to denote the mental activity that occurs during sleep. But this is by no means a rule, and both words are used both ways.
This ngram graphs the occurrence of dreamed and dreamt in English-language works published between 1800 and 2000:
This shows that dreamed has been the preferred form at least since 1800. And indeed, based on examples found in historical Google Books searches, dreamed has always been more common than dreamt, and dreamt is likely the newer form. We find an abundance of instances of dreamed in sources from the 16th and 17th centuries, while dreamt is rare until the 19th.
Though dreamt may be fading out of the language in favor of dreamed, it still appears in some major publications—for example:
Instead, Match Day 33 dreamt up the plot of a manager who leaves the league’s most left-leaning, anti-capitalist club. [Guardian]
In fact, during the night he actually dreamt up a new ring design using ethical diamonds. [New York Times]
For decades, developing countries dreamt of sky-high commodity prices and rock-bottom international interest rates. [Sydney Morning Herald]
But these same publications are more likely to use dreamed in most contexts—for example:
Twelve years previously, when I was pregnant, I had dreamed of bringing him up with my passion for music. [Guardian]
Roach, who will be director of boxing strategy, said he had always dreamed of winning an Olympic medal. [New York Times]
Soccer now is a fully professional game and has a television deal it could have only dreamed of less than a decade ago. [Sydney Morning Herald]