The .308 is essentially a shorter, downloaded .30-06. The -06 is requires a long action reciever where the .308 is a short action. Some people say a short action is more accurate because there is less flexibility due to a smaller ejection port and a shorter bolt. I don't think you can go wrong with either. Most experts (I am not an expert) will agree that the .308 is more accurate at long range but the .30-06 retains more power.
That being said, for my money I'd go with a .30-06 for big game at long range and stick with my .308's for human sized targets at somewhat closer range..