Although both of these phrases are common today, some believe that different than is always incorrect. The reason is easy to understand. The word than follows a comparative adjective or adverb. For example, her car is faster than mine. Or he runs faster than the rest of the team.
But the word different is NOT a comparative adjective. Hence, the use of than is incorrect.
Different from is preferable when two things are being compared to each other.
Your house is different from my house.
If you used different from and then a clause, you would have to introduce the clause with a subordinating conjunction (usually what) that forces the clause to act as a noun, that is, as the object of the preposition from.
Example: Cars today are different than they used to be, but his old car is not different from mine.
This problem is thoroughly discussed in the Parts of Speech section on Adjectives. Click here for the beginning of that discussion.