An explanation owns what happened. An excuse tries to hand the blame to someone else.

The words can be nearly identical; the difference is where the sentence lands. An explanation walks toward you and stops at what you did and what you will do next. An excuse walks away and leaves the responsibility with traffic, a teacher, a phone that died. People can hear which direction it is going within a sentence or two.

Next time you are late or short on something, give the reason and then the repair. One line of context, one line of repair.