I have one 308 WIN, and it is my favorite deer rifle. It is a Savage 99, and I download it to 300 Savage levels.
I have 9 30-06 rifles. Most of the time, I shoot them with a load that is slightly downloaded to 308 WIN levels.
The result in both cases is a fairly dramatic reduction in recoil. I am not adverse to recoil; I'm a walking landform. However, I figure that I need not put up with it when I don't have to. I also am hunting places only 800 feet or so above sea level with less than 200 feet of altitude. A little extra weight is not a problem.
For my taste, I prefer the 30-06 over the 308 for the simple reason that, when loaded to near-MAX levels, the 308 Win seems to have much sharper recoil than the slightly downloaded '06. It is a minor issue, however. I do not a whole lot of difference in the two beyond that.