Hello,
this is how I understand the difference:
I was going to phone you yesterday, but I forgot. - unfulfilled intention, I intended to do something, but it did not happen...
I was about to go out when you called. / I was about to give up when I finally came across a solution. - it is sort of neutral, nothing dramatic, the intended action was somehow interrupted by something else if it is in the past, I think.
at present - I am about to eat my dinner. / She is about to leave for work.
The building was on the verge of collapsing. / She was on the verge of leaving her boyfriend. - sounds more omnious, usually something bad/dramatic was going to happen
be about to / be going to + the infinitive
be on the verge of + -ing / noun
Greetings,
Petra