Most folks replying to this post, are walking right up to this, but not getting it into words: 20 gauge guns are frequently made on a dedicated receivers/frames, which are smaller and lighter than a 12 gauge receiver frame, and the barrel or barrels are lighter. 1 once of shot, fired out of 12 or 20 in the same sized receiver/frame/barrel length, are going to recoil almost exactly the same. That same 1 once load in dedicated 20 gauge receiver/frame (smaller and lighter) is going to kick more. Still that is not a lot of recoil, and the guns are a delight to carry after game all day, the lighter weight appeals to smaller hunters as well as big ones.
A 20 gauge on the correctly sized frame/receiver with 7/8 oz loads is perfect for a person looking to go light, without undue recoil.
Do all the clays and bird/rabbit hunting with a pair of identical 20 gauge doubles (Ithaca/SKB 150 & SKB 280), where power is needed, deer driving and turkeys is for the 12 gauge.
Handload the old 2 3/4 inch 20 gauge baby magnum load which has 1 1/8 oz of shot, which is hard to distinguish in effect from the standard 12 gauge 1 1/8 oz field load. People, really into shotguns will talk junk about the long shot column in heavy 20 gauge loads, saying that puts it way behind the 12, maybe it does on passing birds, but the critters I shoot at are usually directly in front and headed straight away.