There isn't a deer alive that will be able to tell the difference if you hit them, so in terms of which is better for results, it's 6 of one and half a dozen of the other.
The difference is in how you shoot it.
The recoil is noticeably less in a 6.5, so as a new shooter you'll be less likely to develop any sort of flinch and be a little more accurate. If you're hunting longer distances (ex over 200 yards) it has less drop and wind drift. If you're sticking with just deer there's no reason to go any bigger.
Ammo is cheaper in .308 and more selection is available so you'll be able to afford more to practice and get better. At shorter distances it delivers more energy. If you ever want to go after bear, moose, elk it's a little more capable with the bigger animals.
If i were introducing a newbie, of the two i would pick the 6.5 just for the shootability, but to split hairs even more i prefer the 7mm-08. It's the middle ground between the two with better trajectory than the .308, better bullet energy than the 6.5 without much more recoil and just the sweet spot between the two. The downside to the 7-08 is that ammo is a little more expensive and less available.