I have a Tikka T3 Varmint in .308 and a rem 700 in .30-06. It may be like comparing apples and oranges but I find myself shooting the Tikka more, possibly because of the reduced recoil and because the military surplus is pretty cheap. (well was until about a year ago, but I have faith that after the Iraq war is done the surplus market will be ripe again).
As for white tail, both guns are fine. Both have plenty of power an accuracy within hunting distances. The real difference comes with a follow up shot, if needed. The .308 fits in a short action bolt while the -06 is a long action, therefore the time needed to cycle the bolt is longer with the -06. Keep in mind that it may only be an extra half second of cycle time, but it could matter when your hunting. The biggest advantage, and reason for the invention of the .308, is the reduced recoil which is quite noticeable from similar guns, ie same barrel length, similar bullet weights and similar gun weights.
Even with the moderate advantages of the .308, the 30-06 is probably the best bet. It has more than enough power to take white tale and bigger animals because hell, you never know, your buddies may want to hunt elk in Canada next year and you will have a rifle for the hunt already. Buy a good rifle in either caliber and you will be perfectly happy with it, I know i love both my -06 and .308.