The 7mm-08 would give better trajectories with ligher bullets than the .308--but not a lot. The .308 would allow a bit better use of heavier bullets--but not a lot.
Off-hand, I'd say the .308 is a better all-around cartridge. Just not a lot.
I'd venture that the short barrel of a Scout-type rifle would limit one's safe-kill range on larger game like elk. But I don't want to get into that particular argument; some folks never seem to see one inside of 400 yards, and others keep stumbling over them.
I guess the summary is that with 150-grain and 180-grain bullets, the .308 will handle almost any "normal" hunting you're likely to do...
Good shot placement solves a lot of ballistic problems.
FWIW, Art