My pick
Your question calls purely for opinion, because, ballistically speaking, both cartridges will get the job done.
.270 shoots flatter, and I'm a flat trajectory kind of guy. Comes in useful out in the sage brush.
.308 allows you to throw a larger bullet, but for deer, I can't see needing anything more than a 150gr, no matter how big the mulie(or white tail in your case) is.
.308, due to it being a short-action cartridge, also allows you to carry a slightly shorter/lighter rifle, if that is your choice.
That being said, since I just made this choice myself a few weeks ago, I went with .270.
greg