Yes what AXXE said! "Exactly" Both have around 15 Lbs of recoil.
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With all variables equal, such as rifle model, stock, overall weight, etc. the same, the recoil is pretty close. Under these conditions I am not sure I could tell the difference if I did not know what cartridge I was shooting.
In theory the 30/06 and .308 would have the same recoil in the same gun.
Nope, better pull out your recoil calculator, shooting the same weight of bullet with max loads for each you will find the 06 is on average 80-90fps faster and burning about 10-12gr more powder, that makes for a considerable difference in felt recoil. Mine are even more extreme my 30-06 and 308 weight the same, have the same LOP and both have top notch recoil pads on them, but my 06 is shooting 61.5gr of RL19 (Nosler Max) and the 308 is shooting 44.5gr of Varget (pressure limit) the difference in recoil is easily noticeable.