From the Sierra Exterior Balllistics website ... (with my italics and bold):
The 308 is an extremely easy cartridge to reload. Like the 222 Remington and 6mm PPC, the 308 is an extremely accurate cartridge. It probably is the most inherently accurate 30 caliber commercial cartridge ever produced. As a competitive cartridge, the 308 has been used in benchrest, highpower, long-range, silhouette, and 300-meter international matches. Few, if any, other cartridges have been so successful in such a wide range of shooting disciplines. It has also become quite popular as a hunting cartridge and is adequate for most North American big game species, particularly deer-sized game. Though frequently compared to it, the 308 cannot match the performance of the 30-06. The difference between the two, however, is insignificant unless bullets of 180 grains or heavier are discussed.The 30-06’s greater capacity and ability to use slower powders give it an undeniable edge.
Like Bill said ... recoil gets to be the problem when you start to go for the 30-06 advantage. My gut feel is that due to the current popularity of the .308, it means that there are more quality 'competition' parts available for the .308 than the 30-06 (and yes, I shoot a .308 too).