I've had both, liked them both. In my experience, the .270 seemed more accurate overall. In a box-stock Browning A-bolt, it would should 3/4" groups at 100 yards all day long. Every day. No matter what. And with several different types of factory ammo (this is why I think the cartridge itself is accurate, not just an exceptional rifle). With reloads, the first reload I ever tried, it shot 1/2" groups. Without even trying.
Now for the 30.06. It's a great cartridge, no doubt. But I had a Remington 700BDLSSDM (which basically means it's stainless/synthetic, and had a detachable magazine). I meticulously broke in the barrel the same way snipers do. I also had the trigger worked on, the crown recut (it was out of round), and the stock/barrel freefloated. And the best groups I could get were 1 1/4" at 100 yds. And I tried every possible factory load and reloads I could get my hands on. Was the rifle a POS? Or was it that the .06 just wasn't as accurate of a cartridge? We'll never know.
BTW, I no longer own either rifle. Could care less about the Remington. Kick myself every day over the Browning!!!!