You can't go wrong with either. The 308 has a much better selection and a wider range of bullet weights to choose from. For that reason I would choose the 308 for versatility. It also has more action types to choose from as it is a short action. For example, I know of very few lever action guns chambered for a 30-06/270 length action but there are a lot for a 308 length action.
A better comparison might be the 7mm-08 (.308 necked down to 7mm or .284) vs the .270. The biggest single difference is long action with the .270 and short action with the 7mm-08 or .308. The short action rifles can be had in lighter weights, which have long been my preference. Bullet weights will vary a little, the .270 is probably best known for the 115 and 130 grain bullets, although 150 grain are available (long & skinny). The 7mm offers about any weight from 100 grains up to 162 grain pretty commonly, I've seen some 175 grain bullets but they are so long they probably lend themselves better to the .280 or 7mm Mag.
The .308 gives you some even heavier bullets, but if I wanted a .30 caliber I would probably go with something that could shoot a little faster like the .300 Mag or .300 WSM. The 7mm-08 has great velocity and ballistics for that size of cartridge.
They are both very versatile calibers and for deer they would both do great. I prefer the .270.
If you ever decide to try reloading you can go from one end of the gammit to the other. You can load a 90 grain for varmits that sreams out of the barrel at comparable speeds to the smaller varmint calibers, or load a heavy slow 180 grain tip for elk. In my mind the .270 just has a slight edge over the .308.
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