I had a 338-06 for several years. My 30-06 shot 200's to the same speed as 225's from the 338-06. At the muzzle the 338-06 wins. At 150 yards the better BC's of the 30's make it a tie and at any range greater than 150 yards the 30-06 has more energy and drops less. Once I figured this out I sold the 338-06 and bought 200 gr .308 bullets.
I guess it depends on the loads.
I get 2775 fps from my 21" barrel 338-06 with 210 Sciroccos.
My brother gets 2875 fps with his 24" barrel 30-06 and a 180 grain Scirocco.
Even with a 3" less barrel the slower started but heavier 210 Scirocco still have over 1700 FPE at 500 yards if that means anything.
My brothers 30-06 at 500 yards is just over 1600 FPE.
Granted the 210 drops roughly 3.5" more, but the 338-06 is also 3" shorter overall and handier. Had it had a 24" barrel and picked up another 50 fps or so from it 210 would have even that much more energy than the 30-06 with a 180 grain and the difference in drop would be that much less as well.
If it enough than any animal could ever feel a difference....not really IMO, but its not like the 30-06 just walks away from the 338-06 when shooting bullets of similar BCs.