If you read the ballistics on the ammunition you find that the power of a 20 ga is almost that of a 12 ga. However, with shot shells you sacrifice numbers in the patter. As was noted before, 20 ga guns are lighter, and this translates to more felt recoil. Many parents and coaches will recommend 20 ga over 12 ga for young shooters and then beat them up so bad they never want to shoot again.
Take advantage of the wider selection of both guns and ammunition in the 12 ga, enjoy the cheaper ammunition for the 12 ga, learn to shoot it well, and never regret that you have enough gun for any game you hunt with a shotgun. If you get excellent at your sport or hunt you can always downsize to a 20 ga or .410, but there is no reason to start out that way.