For starters: The .308 in factory laodings is loaded to around 55,000 psi. The factory '06 stuff is loaded, typically, to around 50,000 psi. With 150-grain bullets, this equalizes the two.
The .308 cartridge case is more efficient than the '06. It works just fine with a 20" or 22" barrel. The '06 is a slightly overbore case, and works better with 24" or 26" barrels.
Because of the setback of a 180-grain bullet into a .308 case, the decreased room for powder means that '06 velocities cannot be achieved.
So: For one who does not handload, and who doesn't need to use bullets above 165-grains in weight, the .308 is superb. And, with the inexpensive ammo, it's less costly to develop one's shooting skill.
For a handloader who doesn't mind the weight of a long tube, and who tends to want a 180-grain bullet at great effectiveness, the '06 is better.
My own pet rig is a Weatherby Mark V with a 26" barrel. With scope, sling and six rounds of ammo, it weigs 9.5 pounds. I've put in many an eight-hour walking-hunting day with it. Yeah, it's fatiguing, somewhat--but it's a really nice Ma Bell critter. I grant that I don't do all that walking anymore, and have gone to a much lighter weight rifle, but I'm not really a spring chicken, either.
Art