is there any difference between this two words?
The company is about to go bankrupt
The company is on the verge of bankruptcy.
The company is on the verge of going bankrupt.
I am on the verge of leaving
I am about to leave
So about to needs to be followed by a verb, while on the verge of can be followed by a noun or a gerund.
"On the verge of" is commonly found in contexts that suggest "dramatic or alarming or disconcerting closeness to something momentous":
1. I'm on the verge of a nervous breakdown = I'm extremely close to a nervous breakdown.
2. ...on the verge of bankruptcy...
3. ...on the verge of suicide...
4. ...on the verge of asking her to marry me...
"About to" can be used in momentous circumstances, but it doesn't convey the same sense of "drama" or "alarm" or "disconcertingness". In fact, it's more likely to convey a sense of impersonality or misplaced calm:
6. I'm about to suffer a nervous breakdown.
7. Hello, is that the police? Yes, this is MrQ. I thought I'd just call to let you know I'm about to shoot my wife's new boyfriend. Yes, you did hear me correctly...
Or sometimes pleasant surprise:
8. I'm about to become a father!
This is probably because "about to" mostly turns up in neutral contexts:
9. I'm about to call Miss T. Did you want me to ask her about that invoice?