I don't hear "go off" very often used in conjunction with "event" to mean "to happen." What is the difference between
It's amazing that our party went off so well!
and
It's amazing that our party went so well!
Is it a BrE/AmE thing?
A party "going off so well" strikes me as a rather old-fashioned way of speaking. I imagine a rather grand old lady (a dowager duchess, perhaps) saying such a thing. (The word "off" would be pronounced "orf" by such a lady.) The problem with going off is that exactly the same phrase is used in respect of a foodstuff that has become rotten or sour: "the milk has gone off, so you will have to manage without". Best avoided.