I have owned and shot both calibers over the years, and still have one of each. I don't have a lot of experience on big game with either, deer being the largest taken with each caliber, but if I had to choose just one for elk, it would be the .338 mag.
As some have noted, the .338 Win. Mag. is a thumper, especially from the bench when sighting in or doing load development. However, in the field under hunting conditions, the odds are that you won't even remember the recoil.
I'm used to shooting big caliber rifles, so it really doesn't bother me that much, but I realize that some people are more recoil sensitive. You need to choose a caliber that you are comfortable with, even if it's less than a .300 Mag. We all have to consider our limitations and what we can handle effectively, and I think that's more important than the caliber itself, within a given caliber range of course.
The frontal diameter of the .338 over the .300, plus the added bullet weight, will make a better elk cartridge, in my opinion. Having said that, if a .300 Win. Mag. is all I had, I would choose a good bullet and go elk hunting.
You won't go wrong with either cartridge, but the .338 will give you an advantage. It's not considered one of the top choices for Alaska or for African plains game for nothing.