There is little difference ballistically between the '06 and the .308. With heavier bullets and bigger case to reload, the nod would go to the '06. A little milder recoil, and short action, give the nod to the .308!
The .308 is one of my two favorite cartridges! If you don't need heavier bullets than say, 160 gr. the .308 will do anything the '06 will!