Ballistically, the 338 AM (300 gr SMK) and 375 AM (350 gr SMK) in same barrel lengths will produce very similiar velocity. My preffered accuracy loads are around 3250 for the 338 Allen Magnum and 3200 fps for the 375 Allen Magnum. Top end loads give the 338 slighly more velocity, 3400 fps vs. 3300 fps but still very close.
BC between these two bullets is **** near identical. Some rifles will need up to around .820 bc for the 300 gr MK but I generally use .800 for most of my drop charts. The 350 gr SMK is right around the .780 level mentioned by Cinch. Again, pretty **** close.
So whats the difference, more energy for the 375 and longer barrel life but also more recoil. Not bad but you can tell its the 375 when the trigger breaks.
The 338 AM on the other hand is warm and fuzzy to shoot but with a bit shorter barrel life as well.