"I'm going to" announces the intention to do something in the future - at any point in the future. "I'm about to" is more immediate - it means "I'm at the point of" doing something or "I'm going to do something any minute." "Stop making that sound - I'm about to lose my mind!" "I'm about to give you some shocking news - you better sit down." "We were about to sit down to dinner - will you join us? "Just when the train was about to hit the woman, the boy pushed her out of the way and saved her life."

"I am going to do something" = I will do something in the future (but I am not saying exactly when. "I am about to do something." = I will do it right now or very soon. 'Very soon', of course, is relative to the task. If somebody is about to build a house, it might mean within the next few days. If somebody is about to speak, it would mean that they will speak in the next few minutes, at most.