Do you mean the difference between 미안합니다 (mianhamnida) and 죄송합니다 (joesonghamnida)? Hopefully I got it right since comparing both makes more sense. :D Next time can you type or copy paste the Hangul of the Korean words you want to ask about? ^^ Anyway...
Both mean the same thing, asking for apology/saying sorry. The difference is, 죄송합니다 (joesonghamnida) is in formal polite speech (to someone older than you, in a higher social status than you, in formal settings..etc) while 미안합니다 (mianhamnida) is in formal polite speech (to someone the same age as you or younger than you, in lower social status than you, etc.)
But, another comparison too is 미안합니다 or 미안해요 just means "I'm sorry" but 죄송합니다 or 죄송해요 means "I am really/truly sorry" with more emphasis on it.