I own both, there's really not much that hasn't already been said in this thread.
The 7mm is an excellent rifle if you are looking to use one gun for a broader range of game. Mine has killed numerous moose, ranging from smallish-average to truely monsterous bulls. 160 or 175 grains, 50 yards or 250 yards, it hasn't mattered. It is also an excellent rifle for caribou and will be my back-up for Muskox as well.
The .338 Win Mag is a bigger hammer that will do everything the 7mm will do, but might seem a bit overkill for smaller game. I've killed moose with this one too, and it does a great job on bison as well, I wouldn't hesitate to take caribou or deer with it either. I've tried 200, 225, and 250 grain bullets. IMO there's no point in using this calibre unless you're shooting 250 grains. Lighter bullets will do fine, but I'd prefer to shoot a 200 grainer from my 300 Win Mag.
I haven't hunted anywhere that requires a truely big gun to fend off toothy critters, but I suspect that even a 338 will be marginal if Smokey catches you elbow deep in your kill. That being said a .338 packs quite a bit more oomph than the big 7, so I suppose if you hunt in an area where you have the potential to be a meal the .338 is the logical choice.
Recoil? I think that the .338 Win Mag has been maligned by the label of heavy recoil. I'm on my second .338 and in my experience this calibre is easily managed with practice.
7mm Rem Mag, .300 Win Mag, and .338 Win Mag. It seems I've got a bit of redundancy in the locker. Good for me.