The aim of this work was to perform a simulation analysis of the dynamics of a freight wagon with a variant vibration damping: dry friction and viscous damping. The following mathematical models of the damping characteristics are presented: the Maxwell model and the Kolsch model. The differences among the types of damping were first analyzed based on the dynamic responses of the 1 DOF model. Simulation studies were then carried out in a VI-Rail environment with the use of S-curved track models comprising short straight sections connecting the curves. The track models differed in the values of curve radii, cant, and length, which made it possible to run at different speeds. The multibody model of the vehicle represents a typical two-axle freight wagon. The dynamics of the wagon model were investigated for two states: empty and laden. Standard kinematic and dynamic values were compared in order to investigate if the nature of the damping has a significant impact on the dynamic properties of a freight wagon. The analysis of the simulation study showed that replacing dry friction damping with the viscous one can generally reduce forces acting on the wheel–rail contact, which, in turn, can be related to improving the running behavior of wagons while reducing the negative impact on the track.