Do you reload? If so, and this is going to primarily be a deer gun, consider something other than those two cartridges. If you want something that's going to minimize drop, while keep recoil manageable and to a minimum, consider the 6.5mm/.264 caliber chamberings - .260 Remington, 6.5x55mm, 6.5 Creedmoor, 6.5x47, 6.5x284, etc. The .260 Remington can be loaded to duplicate the trajectory of the .300 Win Mag and carries more energy and velocity at 1000 yards than .308 with far less drop, something close to or around 3 feet less.
If you don't reload, the factory ammunition choices for .260 are kind of limited. There were about 4 or 5 choices on the shelf at my local Cabela's when I went in the other night and were pretty much limited to 120 grain or 140 grain. Either will kill deer, but you need to find what works best for your rifle depending on the twist rate, etc. .270 and .308 have far more factory options, but if you want accuracy out that far, you probably should consider reloading, depending on what you consider to be acceptable accuracy.