A banquet is almost certainly a more formal affair than a feast
You could have a feast eating with your fingers, sitting on the floor. Not so a banquet.
Both banquet and feast involve food, an affair might, but not necessarily. Affair is a bit like event.
You could invite people to a wedding feast or a wedding banquet without provoking any strange reaction. Not so a wedding affair
I forgot to mention first that my understanding of these terms is quite at odds with Prometo's - and I think we are both right in our respective contexts.