Generally, the 308 is more efficient, but the 30-06 performs better when using heavy for caliber bullets (190-220 grain). Both are great cartridges, and I believe the reason that most match shooters have changed to the 308 is because it is a wee bit more accurate.
That having been said, between the national match 30-06 Garand I owned, and the national match M1-A I owned, I enjoyed shooting the Garand more.
I think of the two cartridges sort of like I do good motorcycles. The Garand/30-06 is old, and the 308/M14, M1-A is newer..sort of like comparing a Hayabusa and a mint 1954 Harley. Sure, the Hayabusa is faster, but I'll bet if you showed up at a bike rally with that mint 1954 Harley, more guys would be drooling over the Harley than they would the Hayabusa.