Right now I have one .270 and three .308's, and my wife has another .308.
One thing the .308 can do that the .270 can't is shoot the same weight bullets at the same muzzle velocity, at slightly lower pressures with less powder. This results in somewhat less recoil, around 20% with 150-grain bullets according to the basic recoil-energy formula.
Whether not this matters to real he-men is another question, but for lots of practice I vote for the .308.
The .270's smaller-diameter bullets so have slightly higher ballistic coefficients, so in theory will work better at longer ranges. But if you run the numbers for the same brand/type of hunting bullets, it turned out the difference in trajectory and wind-drift at 400 yards will be about an inch, which in real-world big game shooting doesn't matter. Plus, there are more higher-BC bullets available in .30 caliber.
