We investigate theoretically the influence of the long-range and short-range potentials on the plateau structure of the above threshold ionization. In a considerable range of laser parameter, the above threshold ionization spectra of the atoms in the long-range potential always exhibit a clear double-plateau structure; as for the atoms with a short-range potential, the boundary of the double-plateau in photoelectron spectra is no longer clear, and with the decrease of laser intensity, it transits from the double-plateau to the single-plateau gradually. The numerical simulation based on classical analysis and quantum mechanics illustrates that in different model potentials, the distinction of ionization rates as well as the difference of the electronic elastic rescattering cross-sections results in the difference of plateau structures. In addition, the influence of intensity distribution of laser pulse on the phenomenon is discussed.