No easy answer. Lots of variables.
Are you going to shoot paper or large game animals at long range?
Generally the .270 will have a flatter arch than a 30-06 from muzzle to say 400 yards. If you zero both at say 200 yards and then shoot at 100 yards and again at 300 yards (without adjusting sight for range) the .270 will hit closer to point of aim than the 30-06 in the vertical axis at both 100 and 200 yards because normally the .270 has a higher velocity equating to a flatter arch. 32 feet per second per second of bullet fall is a given (gravity). Bullet velocity is the variable that creates the arched path of the bullet over distance. I define accuracy as putting the holes as close together as possible at a given distance.
How much wind is blowing can effect accuracy. Lighter bullets are blown about by the wind more than heavier bullets.
Strive to live up to the opinion that your dog has of you.