I have to agree with two of the thoughts here before I give you my opinion:
First and foremost, get what you decide you want, or need.
Second, both the 300 mag and the 338 mag are overkill for deer or black bear. WAY overkill. Of course there's nothing wrong with that if you just like to shoot big calibers.
As for my own $0.02 worth, there's almost nothing the 300 mag will do that a 30-06 won't do. The benefit in speed and range is the next thing to nothing. The jump up to a 338 makes bigger holes and so is a better choice in my opinion. Of course that's coming from someone who has hunted elk for many years and who still lives in big bear country. I'd still say that a 300 is too similar to the grand old cartridge, the '06.
For what it's worth, I had a 30-06 Ruger 77 MkII All Weather with the black factory stock. It actually had more felt recoil than its replacement, a 77 MkII stainless with a laminate stock, Decelerator pad, and Barnes muzzle brake. I know that it's comparing apples to pears, but that 338 is a wonderful shooting rifle that I have no problem shooting a lot with.