Art, I don't disagree one bit - to a certain extent this is a subjective Ford vs. Chevy discussion since there are legitimate arguments for all the good and thoughtful opinions expressed so far in this thread; and it probably has a lot more to do with personal preference, sentiment, etc. than it does with real-life differences.
While I agree 100% that the .338 Win Mag would be a thoroughly acceptable "one-gun" deal on an Alaskan hunt - it just wouldn't be the one gun I would take.
As an anecdotal experience however, I do have a good friend who hunts a lot in a lot of different places - he bought a brand new .338 Win Mag for his first Alaskan bear hunt, then found when he got to the outfitter's headquarters that they much preferred - as in required under all but the most unique circumstances - their hunters use at least a .375H&H. Obviously he should have checked into it a little further, but he didn't since he's a very busy guy, it was his first Alaskan bear hunt and he tends to act first and worry later anyway. As it turned out, they allowed him to hunt with the .338, but he had to pass a very stringent shooting/accuracy test before they did.
I'm not saying every Alaskan outfitter has that stipulation, but this one did and my friend only goes first class, so they would have been top-tier outfitters.
Again I don't think the OP can go wrong with a .338 Win Mag either.
FWIW I think either rifle will work, but this tends to be a personal preference deal too - there are another dozen rifles (or two or three) in that general price range out there that will do the job just fine. You could spend a bunch more money and not get that much more rifle, but again FWIW, try as many as you can, buy the one that fits you best and spend the extra money on the very highest quality optics you can reasonably afford.