jmurman,
All things being equal there really is no difference in accuracy between the two.
There certainly ARE differences between ammo types, barrel quality, and general rifle "tightness" that have an effect..... but if you put a new barrel on a rifle and shoot good quality milsurp in it you should see decent accuracy whether it's '06 or .308.
The one caveat I can add to this is that, like you said, .308 milsurp is easier to come by and cheaper than 30-06, it's generally much "fresher", i.e. 20 years old or less, while most '06 milsurp is 30+ years old.... so that MAY make a difference in the general quality of available ammo and how accurate it shoots.
Just based on my experiences alone.... the .308 is a bit more accurate... but that's because I'm shooting "newer" ammo in a rifle with a new commercial barrel, as opposed to all my '06 Garands still having the original GI barrels.
To throw more mud in the soup.... Some GI barrels shoot GREAT with good Match quality handloads. This does not quite reach the level of my Match Grade .308 M1, but definitely betters my "plinker" .308 M1 (in which I generally shoot only milsurp).
Just my opinion....
Best to all,
Swampy
Garands forever