I cut my teeth on a .32 WCF. An early Colt SAA first generation. No question about it, a .32-20 beats a .38 S&W and a .38 S&W Special hands down. It was also one of the first cartridges I handloaded. It's also one of the reasons I don't hear so good. Huh01
In terms of performance the 32-20 is better for small stuff than the 38 S&W... no question there.
If you bring the .38 Spl. into the equation it is another matter... There is a large variety of .38 SPL ammo on the market with a wide variety of types and weights of bullets for differing uses. One thing sure, the .38 Spl ammo is going to be a LOT cheaper than 32-20.
I really like the .32-20 idea. As QC says it's not a tough round to load for.
Also, in a pinch you can shoot .32 S&W Longs, Shorts and as I recall .32 ACP from a .32-20 chamber. 75 years ago, it was pretty common practice in Maine for woods hunters to use a .32-20 rifle loaded with High Velocity ("H-V" headstamp) ammo for deer and keep a few .32 Shorts in their pocket for rabbits, partridge or squirrels. (Under No Circumstances should Hi-Vel Ammo be used in revolvers. As CIL used to have a factory in Plattsburgh, you may still encounter this ammo in upstate NY. Hollow point ammo is a dead giveaway....)
Hamilton Bowen has said that the .32-20 was by far the best round going for a "Field Gun".
But as Chuck has said, the .38 Spl. has the advantage of being alot easier to find and procure components for. Factory ammo can be found ranging from low noise flat-faced Target Wadcutters which can work well on edible small game to fast stepping 110 Grn. +P+ JHPs which will really tear up a porcupine.
Lastly, looking at the guns themselves, fixed sight guns from this era have tiny notch and blades.... Colt built many more target sighted .32-20's than S&W. Target sighted .38 Specials are fairly common.