The .338 will always shoot bullets with a better ballistic co-efficient and a higher sectional density than comparable weights in any .35 caliber. The .35 hass the advantage of being able to use .357 acliber pistol bullets for plainking/small gameuse if you are a handloader and are so inclined. Beyond that, I don't know that there is much real world difference.

I just got a .338/06 barrel last month and I'm getting ready to put together a Mauser in .338/06.