Actually there is no difference in recoil. The rifles for .308 are usually enough ligher than an '06 so that they recoil as much or more so than the average '06.
Pure poppycock.
The difference in rifle weight is generally minimal - 1/8 pound in the Remington M700 SPS, ¼ pound in the Savage 11/111. Let’s take the max .308 168g load from Nosler 6, a 2910fps with 52.5g powder, and falsely assume for a moment we can use the same load data to achieve the same velocity in a .30-06, using a .8 pound scope on both cases. Apples to apples. The recoil in the Savage rifle/scope combos (7.55lb/7.8lb) is 22.71lb vs 21.99lb. In the Remington rifle/scope combos (8.05lb/8.175lb) the recoil is 21.30lb vs. 20.98lb. In both cases the .308 Win has slightly more recoil.
In the real world, however, the .30-06 case has about 23% more capacity and more powder is needed to achieve similar velocities. In Nosler 6 the .308 Win max loads use from 43.0-52.2g powder while the .30-06 loads use from 48.0-63.0g powder. A mere 1.5g of powder is all that is required to overcome the weight difference in the Remington’s and 3.0g will overcome the difference in the Savage rifles. If attempting to match velocities the recoil can sometimes be indistinguishable but if trying to maximize velocities in both cartridges the .30-06 recoil is often several pounds heavier.
Is the 100-150 fps you might gain in the '06 really of any real world value? Nope not really. What one will do so will the other. That's really why tho I've long wanted a .280 Remington I have instead always bought 7-08s. I like the rifles the shorter round comes in better so seem to always choose it as the small difference just isn't significant.
With Nosler 6 data, which pushes a 168g AccuBond to 2910fps in a .308 and 3002fps in a .30-06, the .30-06 will deliver the same energy at 345 yards(2095fpe) that the .308 delivers at 300 yards (2088fpe). While that is not important every time, and perhaps only very rarely, only a fool would claim that it is never important.
With some rifles, such as the Remington SPS, both the .308 and .30-06 carry 24” barrels. In other rifles both come with 22” barrels. The difference is often only the length of the action, perhaps ¾”.
One thing I have to say is that I can easily make my .30-06’s shoot like a .308 but I can’t get my .308 to shoot like a .30-06...