I've owned several of both. In equal weght rifles shooting same weight bullets and full power loads I personally could never detect a difference in felt recoil. I tend to prefer short action rifles for the simple fact that you can mount todays shorter tubed scopes more easily for correct eye relief without resorting to butt ugly picatinny rails or offset rings. Never saw any definitive, meaningful accuracy advantage to the .308 in my hunting rifles. All that said, I still prefer the old '06 between the two for it's history and because the .308 is a short, squatty, uninspiring, dumpy looking little cartridge.
