Same level of formality and politeness. Different situations.
죄송합니다 ~ My apologies : a serious expression of regret, admission for some wrongdoing.
미안합니다 ~ (I am) sorry :
(1) apology for some 'not-so-serious' mistakes, embarrassing faux-pas.
(2) expression of empathy, compassion equivalent to "Bad luck!" in English.
(3) close synonym of: 실례합니다 = Excuse me. Pardon me.