Well, for starters, they're two completely different rounds; IMO not even really comparable. The .270 is essentially a necked-down 30-06, which is itself a pretty old cartridge, though still quite capable. A .308 is probably the second most popular centerfire rifle cartridge after .223/5.56; and for good reason - it's very versatile. There is a TON of bullet selection when it comes to 30 cals, yet not nearly as much for .270 (though still a good amount) - If you reload. You can run a .308 on a short action. A .270 requires a long action. Factory loaded match ammo is plentiful for the .308; I'm not sure any is even made for .270, though I could be wrong. The .270 is traditionally a hunting round (some old timers consider it and 30-06 as THE round for deer; I disagree...).
In the end, if you're looking for a basic, entry level cartridge for punching holes in paper at moderate distances, the .308 is your huckleberry. If you want to kill something and make your grandpa happy, go with the .270.
Personally (I hand load for everything), I'd go with something chambered in .243 (I'd have mine reamed to AI specs). You can't convince me that there is a more versatile round out there. You can run anything from ~50gr (for small critter blasting) to 80-90gr. (for medium critter blasting), all the way up to ~108gr. for blasting paper at nasty distances. Plus, it's a 6mm bullet. In case you weren't aware, the 6 and 6.5mm bullets are where it's at when it comes to the distance game. 30 cal could be one of my last choices. But that's me, and I like to do things a bit differently.