I've owned, shot, and reloaded the .308, 7mm08, and .270 for decades. In a same make and model rifle, given a 150gr bullet, I don't think I could tell the difference in recoil.
Out to 300yds, the .270 will shoot a little flatter, the .308 hit a little harder, and the 7mm08 essentially indistinguishable between the other two.
If shooting F class competition, some classes are limited to .308.
Otherwise, the 7mm08 is superior past 300yds.
I'm not a big fan of the .308. I only shoot CMP high power, so my competition rifle is a AR15 A2 Match rifle. The 5.56 has eclipsed the 7.62 for service rifle competition. My .30cal hunting bolt actions are a ultra lite .30/06 and a Savage M110 in .300 RUM.
I guess I'm just biased against the .308 as even my most accurate one wasn't as accurate as all but one of my various .30/06. The Garands excepted, but they aren't any slouches, either...
pick what tickles your fancy or what ever you stumble across the best deal on. All are good choices. The particular make and model rifle will be more significant. Decide first on characteristics and features you want. Then caliber, chambering.