If you handload both cartridges to their potential without overloading, the '06 will best the .308 by about 10-12% with 150 and 165 grain bullets, and roughly 15% with 180s. Move up to 200 grains and the .30-06 really starts to shine.
Bit what does that mean, really? A little more punch and a little more penetration...and a little more range. On the receiving end, the animal will drop to either one if well-placed out to any practical distance. The .30-06 will just do it a bit farther out. On a practical level...with the average sporter, tuned for accuracy fed a load that it likes...you can figure on about another 50-75 yards...mainly because of the flatter trajectory.
There are those who like to take their shots at a great distance, and there are those who won't take those long shots for fear of crippling the animal. I'm in the second camp. I've self-imposed 300 yards, and the conditions really have to be good even for that. 200 yards is a limit more to my liking, and either cartridge will bring home the venison at that distance.