Second class? NO! I doubt that any animal shot with either would know the difference!
The plus in the accuracy edge would be that of short action vs. long action, the plus of the '06 is reloading and heavier bullets!
I don't own an '06, and the .308 is my favorite caliber, if I owned one gun, yes, it would probably be an '06! There is no doubt that the '06 is a great cartridge, but.......
there was once a good article written about the '06, "Jack of all trades, Master of none!" It is a great one-gun cartridge, but anything the '06 can do, something else can do better! For deer, anything the '06 can do, the .308, or .300 Savage can do, WITH less recoil! For larger game, with heavier bullets, anything the '06 can do, magnum .30 caliber, or larger can do better! For long range shooting, the 7MM's and some other mag calibers, can shoot flatter, and hit harder!
Again, I am not knocking the '06, the difference between it an the .308 Winchester ain't worth losing sleep over!