I never noticed a huge difference between 12, 16, or 20 gauge on the species I hunt, those being pheasants, grouse and woodcock. The 12 seems to give me about ten more yards confidence, but, truthfully, on target with a 12 is on target with a 20.
I never noticed any difference on clay targets.
You don't see as many for sale new these days, but, folks on the continent sure loved the 16 gauge for most of the 20th century. In France, when you met a guy with an old shotgun in his house, it was a 16 gauge 99% of the time.
Best,
Ted