I have two .338-06s, so my bias is clear. One is a Ruger 1B custom job with a Krieger barrel and the other is an Interams Mark X Whitworth with a Shilen barrel.
I have been mostly using WBY factory 210gr NP loads but plan to reload the same bullets and TSXs in 225 grain. The former are very accurate in my rifles with surprisingly lively recoil -- not punishing like a .338 WM, just active.
A WBY told mde that the Ruger with a 26 inch barrel should get slightly over 2800 fps with the factory 210s. (I may not be able to approach that with handloads.) If so, that would outstrip the .30-06 and the BC of i.338 NP bullets over similar .30-06 bullets also offers some more advantage in longer range accuracy.
They will be my elk rifles next year.
oneb