The stability of salt cavern storage during prolonged operation is a crucial indicator of its safety. This study focuses on an operational underground gas storage facility, conducting comparative numerical simulations for the storage of natural gas and hydrogen. We investigated the evolution of stability for natural gas and hydrogen storage under long-term storage conditions. The main conclusions are as follows: (1) A new equation for stress equilibrium and constitutive relations are derived. (2) At the same storage pressure, the effective stress at the same position in the interlayer is greater for hydrogen storage compared to natural gas storage, signifying a higher level of danger. (3) At the same storage pressure, the displacement at the cavity top for hydrogen storage is greater than that for natural gas storage. The displacement difference between the two is greatest at 9 MPa, amounting to 0.026 m. (4) Due to hydrogen’s lower dynamic viscosity and higher permeability, the depth and extent of the plastic zones within the interlayers are greater compared to natural gas. When the storage pressure is 15 MPa, the depth of the plastic zone within the interlayer can be up to 2.1 m greater than when storing natural gas, occurring in the third interlayer from the top. These research findings may serve as a valuable reference for determining the operational parameters of on-site salt cavern hydrogen storage facilities.