A reason is a purpose for which something is done, usually with good intent or logic.
"What is your reason for leaving work early?"
"I had a dental appointment."
"The reason I was late was that I got a phone call at the last minute"
An excuse is a request for pardon or forgiveness.
"You're late again! What's your excuse this time?"
"I got a phone call at the last minute."
The difference between the two sentences is that a person who is reliable and puctual has a reason for being late. A person who is always late, never completes work on time, and is fairly unreliable has an excuse for being late.
An excuse can be good or bad but if you use too many, they can only be bad!