We study ultrafast magnetization quenching of ferromagnetic iron following excitation by an optical versus a terahertz pump pulse. While the optical pump (photon energy of 3.1 eV) induces a strongly nonthermal electron distribution, terahertz excitation (4.1 meV) results in a quasithermal perturbation of the electron population. The pump-induced spin and electron dynamics are interrogated by the magneto-optic Kerr effect (MOKE). A deconvolution procedure allows us to push the time resolution down to 130 fs, even though the driving terahertz pulse is about 500 fs long. Remarkably, the MOKE signals exhibit an almost identical time evolution for both optical and terahertz pump pulses, despite the 3 orders of magnitude different number of excited electrons. We are able to quantitatively explain our results using a nonthermal model based on quasielastic spin-flip scattering. It shows that, in the small-perturbation limit, the rate of demagnetization of a metallic ferromagnet is proportional to the excess energy of the electrons, independent of the precise shape of their distribution. Our results reveal that, for simple metallic ferromagnets, the dynamics of ultrafast demagnetization and of the closely related terahertz spin transport do not depend on the pump photon energy.