Jim, I'd have to say that in equal weight rifle shooting similar weight bullets, that the 338RUM has sharper recoil. That 375H/H just seems to be more shove like that kick. I've got a 338RUM in a 11.5# rifle, braked it's a piece of cake, suppressed it's a firm shove, and bare muzzle will just about cross your eyes. Rifle shoots 300gr SMKs. In the 375 Ruger Hawkeye I had, shooting factory 300's was stiff, but not in the same ballpark as the 338. That rifle was probably 10.5lbs loaded with the 21" barrel. It wasn't something I'd plink clay birds with, but it was a good shooting rifle with more performance from the short package that the 375HH.
Just to be clear, though, that in a lighter weight rifle, with no recoil reducing add ons, both rifles will have strong recoil. But when you're down the barrel at a big brownie, you really won't notice either at all. I mean, even if you get scoped and blood is running down your face, you won't know it until someone tells you or it drips on your hand. Those big bears capture and hold all of your attention.