The only time I see the .308 giving up anything to the '06 is when bullets over 180 grains are used and velocity is a concern.
I am a huge 30-06 fan who suggests 308 Win more often than not, but have to point out that I've applied this same logic to why, as an 06 lover, I don't need a 300 Win Mag.
If I ever burn out my 06 barrel I'll likely replace it with a 308 Win.