Here's some actual recoil (free recoil) measurements:
TRG-S + scope (~9.25 lbs), .338 Lapua, 250 gr. Nosler: 43.6 ft-lbs
9.00 lb rifle, 7mm Rem. Mag, 175 gr. Nosler: 23.5 ft-lbs
9.00 lb rifle, .25-06 Rem., 115 gr. Nosler: 13.5 ft-lbs
So, in terms of free recoil energy, the .338 Lapua will kick twice as hard, all else equal. However, things can be fixed-up substantially. A PAST Recoil Shield will reduce recoil a full 50%, bring the .338 Lapua's recoil down to that of a 7mm mag. But there's another trick that's just as good---relaxing during the shot will also reduce recoil substantially, maybe even another 50%. This is because energy equals force times the distance through which it acts and force equals pressure times the area over which it's applied, i.e. E=fd and f=pA. Solving for the pressure, p, which is what you feel, yields p=E/(Ad).
Therefore, allowing your shoulder to roll with the punch (relaxing thereby increasing d) is as effective as increasing the surface area acted upon (the PAST Recoil Shield). Together, they can eliminate 75% percent of felt recoil and you are in .25-06 territory (if you held your shoulder stiff and had no PAST shield). The second recoil reducer (relaxing), may sound hokey, but artillery is still designed to work this way using a vertical sliding breechblock.
You can also add a muzzle brake
, if so inclined...
Also, the stock design/fit has very a large effect.