Not much...
Like someone said, pellets moving at a specific velocity pack equal amounts of energy individually. It's the number of pellets that count.
A 1 oz load from a 12 will have the same energy as a 1 oz load from a 20(or 16 for that matter) but will probably have a more even pattern, given identical choke and velocity. Also, a 1 oz load in most 20s is a vicious kicker.
What I personally wouldn't use a 20 for would be...
Anything requiring steel shot.
Anything pushing range much past 35 yards.
Deer hunting with Buckshot.
Turkey hunting.
And, under certain conditions,all of the above may also be 20 ga territory.
The big advantage of a 12, IMO, is the availability of so many different loads off the shelf.