In terms of pattern, the difference is almost certainly less with modern ammo (plastic wads) than it was in the pre-plastic wad days--but the difference is still there. The larger bore, all else being equal, will always have an advantage, although that advantage isn't all that significant.

If you're talking pheasants in specific, you can take a lot of them with nothing heavier than 1 oz loads, and the pheasant almost certainly doesn't know the difference between a 1 oz load from a 20 and a 1 oz load from a 12. The advantage to the larger bores is that if you need to go to heavier loads--which, on occasion, you do with wild pheasants--you can do so. A 16 that comes from the factory with 2 3/4" chambers should handle 1 1/8 oz loads, and a 12 1 1/4 oz, as well as a 20 handles 1 oz.

On this subject, I just ran across an old American Rifleman article from 1930, by Charles Askins--in which he talks about the 20, with an ounce of 6's, being a reliable duck slayer at 45 yards. I don't know much about ducks (and of course we have to shoot them with nontox these days rather than lead), but I would not want to trust an ounce of 6's on pheasants from ANY gauge past 40 yards, and that would be with at least an IM choke, if not full. And that's also considering modern shells versus what Askins had available.