.308/30.06's are pussy cat recoil. Spend the day putting a few hundred rounds of 8mm Mauser and 7.5 Swiss down range, then the .06 well feel like a rimfire.
Oh heavens! And a Mosin will rip your arm off and leave you bleeding out on the ground with a ragged hole where your shoulder used to be! I knew a guy who fired an M44 carbine once and the 7.62x54R recoil killed him and his entire family!
All mid-level rifle cartridges have roughly the same basic recoil characteristics, and what rifle they're fired out of has more to do with felt recoil than the cartridge itself.
Some stock designs help you deal with recoil (in one shooting position or another) better than others (because of differences in drop, LOP, comb, footprint of the butt, weight, and so on). Steel butt-plates transmit more recoil than cushy recoil pads. Lighter guns absorb less recoil energy (making felt recoil stronger) than heavier guns. And so on.
.308, .30-'06, 7.5 Swiss, 8mm Mauser, 7.62x54R, etc., etc., etc... all more or less equivalent and only if you had the same rifle, (say an 8 lb. Rem 700 with a limbsaver recoil pad) chambered in each one would you be able to objectively FEEL a recoil difference.
And that difference would be mighty slight.