If it were me, it'd be the .308.
Nothing against the 7-08, and I plan to get a 7-08 barrel for my wife's Handi as soon as deer season's over--
But-
I've hunted with a Remington 788 in .308 since I bought it new in 1981-- faced the same choice then, chose the .308 because none of the local shops had dies for the then still-new 7-08. In the past 25 years I've seen the 165 Speer BTSP put the smackdown on Texas and Alabama whitetails with authority, and do so with sub-MOA accuracy in the aging Remington. Not to say the 7-08 would not do the same thing (and I'm going with the 7-08 for a bit lesser recoil, only reason), but there's something about the 165 gr weight in .30 cal that's special, with accuracy, downrange trajectory, and terminal performance, from what I've seen.
You'd do well with either though-- which one are you Jones'n for? Go with that!
Rick