Not much. With factory loads you are looking at about 100 fps more speed with equal bullet weights from the 30-06. In the real world, inside of 200-300 yards nothing you shoot at will ever know the difference. At longer ranges the 30-06 will be effective on game about 50 yards farther away and have very slighlty flatter trajectory.
I handload for both and can see a bit more gains with the 30-06. Generally speaking with handloads I can improve on 30-06 ballistics by about 100 fps over factory, but can only beat 308 factory loads by about 50 fps. So by handloading you can see a slighlty bigger gap in performance.
The 308's 2 biggest advanages are slightly reduced recoil, and the potetial for a smaller, lighter rifle. If I were considering 2 identical size and weight rifles I cannot see any reason to no just get the 30-06. But there are several companies offering scaled down, trimmer lighter offerings in 308. You cannot get a 30-06 rifle that trim and light. My Kimber in 308 is still under 6 lbs including a scope and mounts. A 30-06 including optics under 7 lbs would be hard to put together. Closer to 8 lbs is more typical.