These two are very similar but not interchangeable in all contexts.
on the verge of (something / doing something) = at an extreme, on the edge, if we cross the edge then something will happen
about to (do something) = it is the next thing that will happen, and almost immediately
About to is a little more certain and immediate than on the verge of.
The policeman seemed to be on the verge of losing his temper.
Be careful! I am on the verge of losing my temper.
Be careful! I'm about to lose my temper.
She was on the verge of tears.
We are on the verge of deciding. (not about to)
He is about to play tennis. (not on the verge of)
I was about to turn left when I saw Mary.
I was on the verge of turning left when I saw Mary.
I didn't speak to them because they were about to leave.
I didn't speak to them because they seemed to be on the verge of leaving.