Not a great deal of difference but I prefer the 308 Win. Bullet variety is a little better. If you're shooting commercial ammo, availability is better.
Using same bullet weights in the two, you couldn't measure the recoil diff by feel. And, using the same bullet weights, say 150 gr, you can push em faster in a 308 Win. Cartridges with body/neck reduction over around 28% start ebcoming inefficient. 7mm-08 is at 31%, 308 Win is at 26% .