Given the same weight, stock design, bullet weight, powder type, and velocity, the 308 will kick less.
It is simple physics. Without going into the equations, recoil is a function of bullet weight, powder weight, velocity, and gun weight. Recoil increases with bullet weight, powder weight, and velocity, and decreases with gun weight.
In the same case design (say 270 and 30-06), the larger diameter bullet will take less powder to accelerate it to the given velocity. The smaller case (308) will take even less powder than the 30-06.
That is why, for 150 gr bullets and the same powder, the 270 will kick more than the 30-06, which will kick more than the 308.
As a side note: This also explains why, up to about 165 gr bullets, the 308 will tend to outperform the 30-06. Above 165 gr bullets, the 308 performance is limited by pressure considerations.