IMHO they are "different kinds of recoil". A 3" 12 thumps ya good & solid. A 338 (or any of the "super mags" like a 300 Ultra, 30-378Wby, Lazzeroni etc) is more a high pitched "crack" whereby the percussion of the high speed bullet & muzzle blast "adds" to it all. Plus the rifle is accelerated rearward quicker (especially a light rifle vs a heavier shotgun). But I'd say that's all "perceived" as more than likely I'd guess the tables & physics will show the 3" 12 to produce more rearward thrust.

Now if you want THE worst & make you forget about the above, then get an ol 835 Mossberg, synthetic stocks, 24" barrel, ported turkey tube & stoke it full of 3.5" 2.5oz Hevishot Nitros from "Nitro Ray" in MO! I've been shooting these things for over a decade... Didn't say I enjoy them tho! The ONLY other gun in my 3 dozen+ arsenal that approaches it is my Marlin Guidegun 45-70 with Buffalo Bore ammo...