On the verge of vs About to

There is a difference.

"About to" implies that the progression toward a change is already underway and that the change will, with reasonable certainty, occur shortly (though with the meaning of "shortly" being highly dependent on specific context). If a bridge is "about to collapse" then one expects that, short of Superman intervening, it will collapse.

"On the verge of" means that factors are approaching the "tipping point" where the change will occur, and, if factors continue in that direction, the change is likely to occur. But factors may not continue in the same direction. The bridge may be "on the verge of collapsing", but then the water stops rising and the situation becomes less dire.