As far as deer hunting goes, or any other big game hunting, the 12ga rules the domain, the 20ga might do, I don't know, but as far as I'm concerned, the 16 and 12ga's are the big game slug guns.
Well, I've honestly never deer hunted with a shotgun, I've been a strictly muzzleloader guy until this coming season when I'm hunting with my revolver. But my dad had done a ton of shotgun (party) hunting for Iowa whitetails before he decided to switch to muzzleloaders (see where I get it from!
), and he'll certainly tell you that there are in the ballpark of 60-70 deer that are no longer with us to vouch for the effectiveness of a 20 gauge loaded with the cheap 2 3/4" Remington Sluggers. Whether it's a 20 or a 12, you're still talking about a big chunk of lead heading downrange, but my dad has always sworn by his 20 gauge because it's more accurate than the 12 with Forster style slugs.
Nothing wrong with a 20 gauge, if you respect its small limitations and adjust accordingly.
Mike