We explore characteristics of onset and damping in a thermoacoustic engine (TE) driven by cryogens instead of conventional heat sources above the ambient temperature by a comprehensive study of a self-made standing-wave thermoacoustic engine driven by liquid nitrogen. The experiments verify the feasibility of enhancing the thermoacoustic oscillation at cryogenic temperatures. The onset temperature difference along the stack of the TE significantly decreases, compared with that of a conventional TE driven by high-temperature heat sources. The resonance frequency of the cryogen-driven TE is smaller than that of the heat-source-driven TE, mainly due to the lower average temperature of the working gas. Experiments and calculations show that the temperature discrepancy between onset and damping is partly caused by the linear temperature distribution along the stack before damping, together with the nonlinear distribution before onset. These results will contribute to a better understanding of thermoacoustic oscillation and to the recovery of the cold energy of cryogens.