Well I think the .308 is more accurate....has been used with some success in 1000yd competitions......

but now which has more energy at 600yds?

Also if you're considering building a long range rifle, I'm not sure either of those 2 calibers would be my first choice.....

may consider a 6.5-284.......but if you long range/accuracy/commercially available for hunting is your goal.....how about a .270? very accurate, flat shooting, plenty of power at 600yds. Or how about a .270WSM....that should be very accurate, beltless, shorter stack....in a quality setup I'm sure it would be.

Or a .300 WSM with 150gr.....though the .270 with 130 or 150 I'd probably take for what you're saying.

But of the 2 you listed, if punching paper is #1, the .308 should be more accurate, but with hunting in mind I'd look at the tables to compare energies, I'm not sure how they compare at 600yds, if the .308 has 500lbs, and the '06 has 1200, I'd be more comfy with the '06 on a deer at 600yds, but of course you have to hit it.....first, so, take that as you will.

Also consider a 7mm-08?