Personally I used a 300winmag for decades as my elk rifle in the mountains, wonderful calibre with true, reach out and kill it capability; very, "sharp" kick compared to the 338 IMHO. I originally went 300 as I figured the 338 might be, "too much"(bought my 300 when 19). Personally after buying a 338 I'd say the opposite. Again, personally, the 300 isn't bad, but it's, "sharp" compared to the 338 if that makes any sense.
As a qualifier, recoil doesn't seem to phase me much but a 300 target shooting can make my shoulder sore if you don't hold it right, "once" where the 338 and stuff like my 450 marlin doesn't seem to ever have that effect.
Could be just me, too, or maybe I'm just getting old.