I can get 190's and 200 gr SMK to higher velocities in the 30-06 than the 308 Win.
That might make a difference in a hunting rifle.
As a target rifle I was talking to friend who is a Wimbleton Cup winner. He went through a 30-06 phase and told me that while he could and did clean the 600 yard target (ten ring 2 MOA) his X count was always 9 or 10X, could not get it higher. He has shot some awesome high X count cleans with a 308.
He had not shot 30-06 at 1000 yards and expressed the opinion that the extra 100 to 150 fps you get with a 30-06 might give a better score, but since we have not done the comparision, we don't know.
If it takes a Nationally ranked individual to see differences between the cartridges, I would not worry about which cartridge I am using, as I won't see any differences.
If I'm not shooting, I'm reloading.