People have been arguing this since way before most of us were born. It wa probably the first published "caliber war" of the smokeless powder era for rifle cartridges.
To the velocity crowd they dismiss the fact that there are more bullet weights available to the '06 as being insignificant, I don't. A deer hit with a proper bullet from either doesn't know the difference obviously, but the ability of the '0 to go up to a 200 gn bullet (or heavier) can't be ignored. Taking down a tough critter under adverse conditions, gives the "06 an advantage and yet it can go down to a 120-130 gn bullet when that is the proper choice.
Fanboys of both will argue for years but I grew up watching Keith and O'Connor go over it and the experience I gained in my youth told me that the .30 cal would be my choice.
Both due best with handloads IMO But there is much more flexibility to be gained with the .30. I picked up, or rather my wife picked up a .270 over a year ago (Remington 760) and while it may just be stock design, I find the .270 with a 150 gn bullet (what I would use for elk) distinctly less friendly to my shoulder than a 165 gn 06 load.
O'Conner made a career and a living out of championing the .270 against all arguments but if he had handloaded a 30-06 round to .270 spacs with the same weight bullets, nothing he shot would have known the difference.