Theoretically, the .308 has an ever so slight advantage due to the decreased case volume (the powder does not move around much in a .308 case; more modern powders allow less to be used for similar performance. In the .30-'06 case, the powder has lots of room to lay a bit differently on each shot fired). However, in the DCM/CMP Service Rifle course of fire, such "differences" evaporate for most shooters. However, at long range (e.g., 1000 yards) the .30-06 catridge offers a real advantage, though, not remarkably.
On the other hand, the .30-'06 cartridge has the wonderful ability to offer a little bit more power at reduced chamber pressure levels. Though it seems odd that the .30-06 has less chamber pressures than the .308 and yet a bit more "power," it's nonetheless true. And, since it's the original caliber, it works!
The real difference is cost & availability of ammo. The .308 will be easier to find, and less expensive. However, you may think a long action should be fed "long ammo"!
In the final analysis, it's really a personal choice. Either selction will be the "right one."Converting a .30-'06 Garand to .308 does require a barrel change, and a slight modification to the ejector spring (not essential but a prudent thing to do). A spacer block in the mag well (to prevent loading .30-06 cartridges) is a really good idea.