A reason is an explanation. An excuse is an attempt to shift responsibility.
Many people will create a disingenuous reason to absolve themselves of responsibility.
For instance, if someone sleeps in and leaves home 15 minutes late in the morning and arrives to work late, they may honestly say, "traffic was terrible on highway 7." And while it's true that if traffic had magically been nice that day they'd have made it on time, the honest reason they're late is because they slept in. The traffic is their excuse.
While true, I think what the OP in the image is trying to say is that even if I give you a reason you say it's an excuse when it wasn't.
I think I'm neruotypical, but I'd get this all the time from my father. I'm not making an excuse, I'm not spinning anything to shift blame, I'm answering the question and their assumption is that I'm lying to shift blame.
Really the conversation in the image should be: why are you an asshole that can't accept that shit happens. Like the following:
"why were you late?"
"I left on time, was walking down the hall, tripped and spilled something so I cleaned it up."
"I don't want your excuses!"
...well I don't know how else to answer your question without simply explaining the facts of the situation...
Most people don't. The issue is a lot of people ask the question, but have an answer already in their mind before they even asked. They don't actually want an answer, they want to trap you and make you feel bad.
I just dealt with that yesterday to make sure my neighbors understood why my dog was making noise (super bad separation anxiety) and not that I thought any of that excuses howling and screeching.