What is the difference between 迷惑をかけた and お騒がせしました?

Question

What is the difference between ご迷惑{めいわく}をかけた and お騒がせ{おさわがせ}しました?

They both seem to mean that one has caused inconvenience.

Answer 1

迷惑をかけた or ご迷惑をお掛けしました sounds really you are apologizing. "I am sorry for bothering (the trouble upon) you."

お騒がせしました has a soft way of begging pardon, "Sorry about the mess/noise."

Answer 2

騒ぐ originally means "to make a noise", "to rant". So you can only use お騒がせした when you "made a fuss". For example, if someone didn't reply to an important email for a long time, that thing itself would be 迷惑をかけた but not お騒がせした. But if his/her laziness visibly caused a lot of trouble, it could be called お騒がせした.

ご迷惑をおかけしました and お騒がせしました both work as an implicit and indirect apology, but usually it's better to add 申し訳ありません or something similar.