Those two sentences have different meanings.
"I'm supposed to go to the meeting" means that I have some kind of obligation to go, it's part of my job duties to attend meetings, I told someone before that I was going to go, other people expect me to go.
"I'm about to" means that I'm going to do something very soon, in fact it's so soon that saying "I'm about to do something next month/next week/tomorrow" sounds too strange. When I say "I'm about to go," it means I'm getting ready to walk out the door within a couple minutes or I'm already on my way out the door. It's immediate and will happen within moments.
(not a teacher, just a language lover)
Student or Learner