It has been Scientifically Proven that they are essentially ballistic twins right here on this forum.
I have carried a 7mm-08 elk hunting and it did feel like an elk rifle in my hands.... albeit, about as light an elk rifle as *I* have any interest in carrying.
Where the .308 & 7-08 are not twins is in the realms of bullets weight and diameter. Since neither is a cartridge that will excite the "speed kills" crowd, I do think the .308 has an edge here. It will, quite simply, put a bigger, heavier bullet into an elk. Try as I might, with all else so damn equal between the two, I can't see how that doesn't give the .308 the edge here.
I'd go .308 and set it up like Brad's- 165-gn medium hard bullet.
In the 7-08 I ran a 150-NPT, and that's what I'd do again, or a 150-NAB.
That's just me. This is one of those "You take Sally, I'll take Sue... there ain't no difference between the two..." things.
(bonus points for naming that song!)