Excellent points above. One important question: do you handload?
As much as I love my .308s (I shoot more .308s each year than all others combined because I'm a competitive shooter), I'd have to say that the .30-06 runs away if you use modern handloading components.
For example, I've got a hunting load with my .30-06 that uses a heavy charge of Vihtavuori N560 to push a Barnes 168-grain TSX out around 3,000 fps. That's edging mighty close to .300 magnum territory, and my rifle loves the load. There's no way you could load to this level in a .308.
As stated above, the reason for this difference is that the .308 is a much more modern design. Therefore, it's already loaded up to its maximum potential in factory ammo. The .30-06, on the other hand, is loaded by the factories so that it is safe to work in old 100-year-old military rifles and Garands. However, once you stuff that long .30-06 case full of modern, double-base magnum powders like Reloder 22 and N560, and use 165-grain or heavier bullets, the .30-06 really takes off, and in a good rifle, it can be accurate enough for serious target use, though, statistically, the .308 is a little more inherently accurate.
Personally, for long-range hunting, I've got a Sako .300 Winchester Magnum topped with a 3.5-10x scope with a mildot reticle, and my handload sends a 180-grain Nosler Accubond at 3,050. I also carry a laser rangefinder and I'm an experienced long-range competitive shooter. Nobody would disagree that this combination is superior to any .308 or .30-06 past 300 yards... but I've never had to shoot that far yet.
If you're really serious about doing this kind of shooting, and you have the skill and experience to do it (only you can say this honestly), then I'd say go with a .300 Magnum. However, using a magnum creates a whole host of other issues, not the least of which is increased recoil, which really makes accurate shooting a question of focus and mind over matter (because recoil effects are cumulative, like lifting a dumbell -- the more reps you do in one session, the more your technique starts to fall apart as fatigue sets in).
At the end of the day, the .30-06 really is the choice to beat, especially if you handload. If you don't handload, then it's a toss-up because in factory loads, the .308 and '06 are pretty similarly matched. However, the .300 Winchester Magnum (or WSM) is head and shoulders above both of these -- as long as you don't mind the recoil, blast, cost, shorter barrel life, and increased barrel heating (which means longer waits between shots in a hunting weight rifle).