Dago,
The 06 is the parent cartridge to the .308. They are the same projectile, same diameter, weight, taper,and construction. They are the same .308 diameter projectile, period. The .308 is simply a shortened 30-06 which means it shoots the same bullet but not as fast.
I said the 06 was superior balistically to the .308 because the projectile in the 06 is simply faster and carries more velocity with flatter trajectory and more downrange impact than the .308. That is the basis for the balistics superiority.
I would assume you would agree that the 300 Winchester Magnum is ballistically superior to either the .308 or the 30-06. It also uses the exact same bullet. Why is it superior? Because it shoots the projectile faster and flatter with more energy down range same as the situation with the 06 over the .308.
I like the .308, I have 2 of them including one set for silhouette shooting. It is still ballistically inferior to the 30-06 Garand I have even if the silhouette rifle is more accurate due to bedding and carefull fitting not to mention a bolt action vs a gas operated old tired Korean surplus battle rifle.
Now go back to my post earlier. Note that I said this: "Using the same care in reloading, the 06 is just as accurate as the .308 is." I did not say more accurate but just as accurate, given a rifle capable of equal accuracy.
As to your question about the 06 being better in any category, go look at some references for larger game requiring more penetration and a heavier bullet. Because of the larger capacity of the case the 06 is better for the heavier bullets 180 to 220 grains that give the deeper penetration through bone and gristle for the larger game. I'm not talking about small white tail deer, but the large brown bears and moose. If you need a heavy bullet at a long range the 06 is superior to the .308. If you need a flatter trajectory with any 308 projectile the 06 gives it to you over the 7.62 nato cartridge. If you need even a flatter trajectory with more energy than the 06 go with the 300 Win Mag.
Now take a chill pill, the .308 is a darn good cartridge but it is still a 30-06 short.