I'm afraid the answer is "it depends". As a small list, it depends on:
- weight of each rifle
- stock design of each and how well it fits you
- loads used in each (bullet weight, powder charge, burn rate, etc.)
But you probably already know that. I'll say that I've never found the .338 Win Mag to be particularly objectionable and that I've shot .300 Win Mags that were less pleasant to shoot than some .338's. In some of these I think stock design made the .300 feel sharper vs. the strong push of the .338 but that's VERY subjective.
For most uses in NA, I'd say pick the one you like, work up a good load (or find the factory load your gun likes with an appropriate bullet) and go hunting. You have a very versatile rig with either with the nod going a bit to the .338 for the really large stuff (but only if you can hit with it just as well). Put a good bullet from either in the right spot and the animal will die.
