In a high end carefully made sniper rifle, the .308 is alot more accurate than the 30-06, in a typical mass produced hunting rifle, you will probably never see the difference.

When you are talking about accuracy, flat trajectory does not necesarily equal accuracy. Some of the most accurate ultra long range rifles in the world are chambered in .50 BMG, which fire projectiles with about the same velocity as a .308. What extra velocity does do, will all else being equal, is decreases wind drift and drop.
The 30-06 only give you about 100 fps advantage in velocity over the .308. For the effective hunting range of either cartrtridge this means very little in difference in trajectory or ability to kill.

If you know the 'come ups' for your sights for a given range and know the range at which you are shooting, extra velocity doesn't matter much.