Not quite "six of one, half dozen of the other". I have both, shoot the 308 in High power matches, used to shoot 30-06, it is plenty accurate just kicks more.
The 308 works just fine up to 175 grain bullets. Then it starts getting pokey compared to the 30-06 with the heavier bullet weights. Both are very accurate, even in a match rifle could not determine if there was much of a difference, you can always get 150 fps more velocity in a 30-06. Powders that work well in the 30-06 also work well in the 308. Old standby's are IMR 4064, IMR 4895, and in a 30-06 bolt, IMR 4350.
Since the 308 was designed to give pre WWII 30-06 ball performance in a smaller package, there is not that much of a difference between the two. But I think the 30-06 is the better hunting round due to a little more velocity and you can load a little heavier bullets at a decent velocity.