In probably 99.99% of cases, there is absolutely no difference between the two. Both of them are usually linked to the exact same bit of code internally in the program.
It IS possible for someone to override this, but virtually no one ever does, because there's no good reason to.
About the only time they have any functional difference is when some program is not responding. Usually you can click on the "X" button and it will bring up a dialog box asking if you want to force the program to close. That's really about it.
It sounds to me like you've run into someone who has a very narrow comfort range, and that involves doing things in a very specific way, whether or not it really makes any sense. They could have a mild obsessive compulsive disorder as well, and if that's the case, you can argue with them all you like, it will never make any difference. It doesn't matter if you're right, or have logic on your side, you will always be wrong in their mind. Easiest to just do it how they want when they're around.