Jesus christ, the 308 Win would do, and FOR ALL INTENTS AND PURPOSES, you wouldnt see the difference on even the largest game between the two..
a 200 gr accubond, partition, X and so on will do anything that the 338 will do on game... Even on a big moose, or elk (hell, Id put money that if you put two side by side behind either they would both die...
Its not magic, nor does it have any special spells that it casts on game to make it work any better...
If you really need to decide between the two, neither are big enough, so get a 50 BMG, or 30mm cannon...
At the end of the day, the most important question needing answered is "which will I shoot comfortably out to my maximum range and be confident in" the answer to that will be more in the rifle your buddy buys than the caliber choice...
I guess a 338 sounds cooler than a 300 win mag... if thats important, get that...