For marksmanship, the answer is: it depends.
What kind of marksmanship do you mean? Do you mean bench rest shooting at 1,000 yards or do you mean accurately hitting your deer/bear/elk at no more than 200 yards?
As for long range accuracy - either are VERY accurate. There are certain aspects of bench rest shooting that make the 308 a preferable 30 cal round to the 30-06. Both the 30-06 (in the famed Winchester Model 70 rifle - see Carlos Hathcock) and the 308 (in the equally famed Remington 700) were used as Marine Corps sniper rifles in the Korea and Viet Nam eras. Today the 308 (7.62 NATO technically) is still being used in the US Army for sniping - and 1,000 yard shots with 30 caliber rifles are definitely not off the table with either 30-06 or 308.
The 308 is technically the more accurate round than the 30-06. There were tons and tons of tests done on this by the military and despite the fact that White Feather had many confirmed kills at extreme distances and kept his Winchester model 70 zeroed at 700 yards, the 308 seems to actually do better for accuracy. Additionally, modern day long range accuracy shooting competitors use the 308 over the 30-06. Unless you are going to be a competition shooter or get into bench rest shooting, or super long range varmint (coyote) shooting either gun is functionally as accurate as you need it to be.
Now, as for the 30-06 - you can achieve one minute of angle (MOA) accuracy with it which means that group sizes are one inch per hundred yards, out to about 700 yards, but at 1,000 yards you wind up with less than 1 MOA accuracy - 14-20 inch groups. Meanwhile with a 308 you can achieve half minute of angle accuracy or even better out to 700 yards (~3 inch groups) and are at sub minute of angle accuracy at 1000 yards (7 to 8 inch groups).
Whoever said that a 308 would not make a big game kill at a few hundred yards is simply incorrect. The question of killing power is about 2 things: energy upon contact with the target (which is a product of bullet weight and velocity) and energy dispersed into the target (which is about the bullet penetration and expansion characteristics). Given that both the 30-06 and 308 use identical bullets themselves, the only variation then is the amount of powder you can put behind the bullet in the case.
The 30-06 has a bigger case so you would think it can deliver a bigger payload in terms of energy and velocity at longer ranges, and in truth it can, but that payload comes at some possibly considerable cost to accuracy. At shorter ranges, the actual advantage goes to the 308 in terms of energy delivered to the target - and of course it is more accurate. In either case, the energy delivered to the target at 250 yards is more than enough to drop any big game in North America - as with anything though, shot placement is the key and while 308 or 30-06 will kill a grizzly or a polar bear - a 300 Win mag or 338 Lapua mag or a 375 H&H mag will kill those dangerous animals more reliably. If you're going to go for dangerous game like that, imo, use a magnum caliber. You want to kill an elk out to 700 yards, 308 or 30-06 is plenty fine.
It has been said that the 30-06 can take any big game in North America with the right bullets - and that's true - but you might not be able to buy those bullets handily - but none of that issue exists if you reload your own bullets - which is mandatory if you want to shoot in accuracy competitions and all but mandatory if you wan to do 300 yard plus varmint shooting.
The 30-06 has lots of handload data for a wider variety of hunting loads than the 308 but there are quite a few for the 308 too. It's a reasonably accurate statement that the 30-06 is useful through a wider range of applications but that when you get within the range of usefulness of the 308, the 308 is superior to the 30-06. Confused yet?
Finally, there is a consideration of weight and function.
The 308 has a shorter bolt throw - this means you have to move the bolt handle a shorter distance to eject and rechamber a new round - that's because the cartridge itself is shorter than the 30-06. This also means that the 308 rifle has less material surrounding the receiver, which makes it lighter as well than the 30-06. That weight difference may be only a half pound, however, the ammo itself is also lighter than 30-06 - less brass in the shell casing.
So, shorter bolt throw means quicker repeat firing, and lighter weight means more comfortable to carry on a long day of many miles of hiking up and down mountains hunting. Yeah, 8 or 12 ounces may not seem like much, but lug it around for a good 8 or 10 miles up and down mountains in the woods and then tell me which you'd rather carry.
All in all, if I had to choose only ONE rifle for all North American Big Game hunting, it would be the 30-06. 600 yard plus shots are just not frequent enough to care that much about imo.
If I were to choose a rifle for ALL AROUND use between those two - such as big game hunting AND defending my homestead from zombies and raiders in a post zombie apocalypse world - it would be the 308 - actually the Ruger Gunsite Scout - google it.
My "go-to" Whitetail deer rifle is neither the 308 nor the 30-06 though - it's a 243. I came to a conclusion a while ago about this topic - all the "if you had to have only one gun" discussions are fun and all that, but who the hell wants to have only ONE GUN ffs? Not me - I want tons of them!
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If ignorance were painful, half the posters here would be on morphine drips.
Everyone playing WoW knows everything about playing two classes: 1) their own and 2) Hunters
09-11-2001 A Day that will be remembered 
