"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.