Both are extremely accurate calibers and very well suited for game up to deer, sheep, and antelope. It's when you start hunting larger boned game that the differences start to show up.
The 7mm-08 will limit your choice of bullets best suited for large game unless you handload. I could not find a factory load over 140 gr. Winchester factory ammo does not list arecommended 7mm-08load for large game. Federal loads 140 gr Nosler Partitions and Barnes TSX bullet thatwould work better for large game, although still a bit on the light side. But with the .308, you can get factory ammo with 180 gr bullets of stout construction that are better for big boned critters like elk.
Also,in factory loads, the 308 will outperform the 7mm-08 when comparingsimilar bullets and weights.The two have very few common weight bullets. Unless you handload, a 7mm-08 will most likely only have a 140 gr bullet max. And .308 will usually start at 150 and go up to 180 or more.
Below is ballistic info from the Federal and Winchester websites The higher velocities and energys are in BOLD
Federal load using Nosler Ballistic Tip:
7mm-08 has a muzzle velocity of 2800
and energy of 2437
for the 140 gr load.At 400 yards, it is at 2094
fps and 1368
.308 has a muzzle velocity of 2820
and energy of 2648
for the 150 gr load.At 400 yards, it is at 2037
fps and 1382
Winchester load using the Ballistic Silvertip:
7mm-08 with 140 gr: Muzzle velocity 2770
Muzzle energy 2386
400 yards- MV 2026
.308 with 150 gr: Muzzle velocity 2810
Muzzle energy 2629
400 yards- MV 2028
So in both cases at the muzzle, the .308 has higher velocity and more energy with a slightly heavier bullet.At 400 yards,down rangevelocity issplit but the .308 still retains more energy.
(I just used these two loads as examples and not as suggested bullets for hunting elk as they are too lightly constructed bullets.However, the Nosler Ballistic Tip in 150 gr is my favorite for both my 30.06 and my 300 Win Mag for deer and antelope. For Elk, I use handloaded Barnes XBT in 180 gr with my 300 mag and factory Winchester 180 gr Fail Safe in my 30.06.)
Both calibers will kill elk with proper shot placement and a good bullet. I would still go for the .308 for afew other reasons.
1)Recoil difference isn't enough to make a difference in your ability to shoot it well.You will never tell the difference in the field.
2).308 ammo is always easier to find. And it is more often on sale.
3).308 will let you go with a 180 gr bullet that is better suited for game the size of elk.
4) If you start reloading, there isn't a caliber out there with a wider range of bullet weights, designs, and purposes than for the .30 cal. Bullets from 100 gr to 220 gr are available.
5) Both rifles, if of the same make, model, and features other than caliber will weigh the same.
Just my opinion and you know what they say about those....