The one advantage of the 20 gauge is it's available in lighter, more nimble shotguns, that are nicer to carry for upland game birds and jumping ducks.
The 12 gauge, generally, is going to come in a heavier shotgun that's not quite as nimble, but ammo in general is cheaper and you can buy 2.75" 12 gauge shells that have a light load and powder charge for small birds. The 12 gauge, though, lets you use heavy 3" or 3.5" shells, depending on the gun, for things like turkeys or pass shooting geese...so it's more versatile. If I were limited to one shotgun I'd keep a 12 gauge. I do have an Ithaca M37 in 16 gauge that would strike a nice balance between the two, but commercial 16 gauge ammo these days is harder to find, more expensive, and loaded light.
If you loaded your own it would make for a nice option...