I have both of these calibers.
.300 win mag ammo costs less if you're buying factory, and it also shoots flatter. Factory ammo is very easy to find. But, I've never shot a .300 win mag that did not slam me in the shoulder more abruptly than any .338 I've ever shot.
.338 win mag throws bullets that are a whole lot heavier, carry energy farther, and penetrate better than a 375 H&H. Factory ammo costs more, and is not as common as the .300 factory.
I consider these cartridges quite different, actually. I personally consider the 300 win as simply a longer range 30-06, and not much more in terms of killing power. I consider the 338 win mag very, very similar to the 375 H&H in terms of killing power and performance on big game.
Really, all these magnums will kill whatever the heck you want, but I'd take the 338 win mag over the 300 for a pure big game rifle.