The correct phrasing is "different from." I can't recall ever using or hearing "different with" and using it may confuse who you're talking to. For example: Different from = My opinion is not the same as your opinion. Different with = My opinion and your opinion are not the same as the other opinions. (This sentence does not clarify whether my opinion and your opinion are different. It only implies that my opinion and your opinion are different from other people's opinions.) I hope this helps!