I've shot and hunted with both...For a few shots,neither one will kill you;but if you shoot a great deal,you will be effected,sooner or later. I have never felt the recoil of either when hunting,but good shooting habits are acquired on the range before you hunt,not in the field.
Recoil is most certainly not just mental; it has physical ramifications as well....
I spoke with an African pro a few weeks back and naturally the subject turned to rifles....he told me a lot of clients bring 300's and 338's but very few could really shoot them well.....even if they thought they could....