There is no difference due to the fact that they both use the same parent case.
The .270 is a little faster and uses a longer bullet to make the same weight as the '06.
The '06 can use larger bullets - ex. 180 gr or 220 gr
The 270 is a little better in the open and the '06 is a little better on larger game.
They both go bang and they both shoot a bullet and the animal that you are shooting at dies and that is the end of the hunt!
I have killed so many whitetail deer with the 270 / '06 that I got tired of shooting and gutting an dragging and actually went out and bought other calibers - just to see what they would do and if they would do something better than those two.
The truth of the matter is - you can't beat a 270 / '06 for hunting less than 300 yards.