To build on the above, the same distinction exists in English.
It's the difference between 'sorry (죄송하다, 미안하다) and 'apologize' (사과하다)
To apologize is an action while sorry is a state of being. 'Saying sorry' is the act of apologizing. An apology is the act of displaying that you feel sorry.
Of course, we often naturally shorten these expressions to a simple "Sorry" the statement itself is "I am sorry" - a description of being. As such, just as Hailey showed in Korean, is again true in English. Sorry can't be conjugated as a verb (I sorryed, I am sorrying) while apologize can (I apologized. I am apologizing.)
The usage in Korean may be confusing as they can both be used with 하다, just keep in mind that