i have both. although i have been hunting for almost 40 years, i just bought an '06 last year and have yet to get a deer with it. the 270 on the other hand, i have owned since 1982 and have taken a truckload of deer with it. the 270 with 130 corelokts is serious deer medicine.
I own a .270. I've shot 23 deer with it since 1996. Nothing wrong with a 30.06 either. Just a matter of personal preference. The nice thing about an .06 is that you go to 180 or larger grain bullets for larger game. That having been said, I think the .270 is the perfect whitetail deer cartridge.
I have owned both for over 50 years. Actually I have several of each. I use the .270 with 130gr CoreLokts for deer but prefer the 30-06 for elk using 150 or 180gr CoreLokts. I have shot three elk with the .270 but prefer 30 caliber elk. Both are great calibers.