Over the past two decades, studies of the dynamics and efficiency of Nonlinear Energy Sinks (NESs), which were proposed as a development of Tuned Mass Dampers (TMDs), have become widespread. The aim of this work is to compare the efficiency and ability to maintain the tuning of a single-sided vibro-impact NES (SSVI NES), i.e., a vibro-impact damper, and TMD, i.e., a linear damper, under periodic excitation. The research was performed using numerical methods; the impact was simulated by Hertz’s nonlinear contact force. Expressive graphs and tables successfully confirm the findings formulated. Both the SSVI NES and TMD maintain their tuning and demonstrate high efficiency across a fairly wide range of structural parameters; however, in some cases, the TMD maintains its tuning better. At the same time, the zones of bilateral impacts for the SSVI NES with optimized parameters are quite narrow and located near the resonance. These phenomena were verified for lighter dampers with a mass ratio of 2% and for heavier dampers with a mass ratio of 6% when several structure parameters were changed, namely, the damping of the primary structure, its stiffness, and the intensity of the exciting force. The SSVI NESs with lower mass ratios exhibit peculiar behavior; some of their parameters take on non-standard, unusual values. Thus, with periodic excitation, TMDs are not inferior in efficiency to SSVI NESs, which always demonstrate complex dynamics and present difficulties in selecting the optimal design.