Nuclear inelastic-scattering studies were performed to infer temperature evolution of iron atom dynamics in magnetite samples exhibiting Verwey transition of first- and second-order (type-I and type-II materials). The possible difference in this evolution could rationalize the distinct properties of these classes of materials observed in heat capacity and diffuse scattering below the Verwey transition temperature TV and could explain the change in transition order triggered by a minute (below 0.3%) altering of the iron sublattice. Although we have found the apparent stiffening of the phonon iron spectrum in the low-temperature phase, at the same time, we have shown that these spectra are rather similar for type-I and type-II materials, rendering the lattice vibration-based explanation of the distinct behavior of heat capacities very improbable. The calculation of phonon spectra, aimed at tracing the origin of various features in the phonon density of states (DOS), has shown that the local Coulomb interaction U may have a large effect on phonon DOS. However, the change in the U parameter cannot explain the difference in heat-capacity results for both classes of materials. Thus, an additional factor that differentiates these materials and possibly is responsible for the discontinuous character of the Verwey transition in stoichiometric magnetite still must be found.