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There is a long formula for free recoil, (which I do not know, but I know it involves bullet weight, powder weight, velocity and rifle weight), but generally at the same muzzle energy heavier bullets kick more.
Given a rifle with equal weight the .338 lapua will kick harder.
A 300 RUM is more than likely in a light hunting rifle while the lapua is more than likely in a heavy target rifle. In that case the 300 RUM will have more recoil.