In order to investigate the influence of structures of substrates on the dynamic properties of a discrete growth model, the restricted solid-on-solid model for Koch lattice and Koch curve fractal substrates, which have different fractal dimensions and spectrum dimensions but the same walk dimensions, is studied by means of numerical simulations. Surface width and distribution of the extremal height of the saturated surface are calculated. Results show that the random walk exponent plays the determinative part in the saturated regime. Although the fractal substrates have different fractal dimensions and spectral dimensions, the value of roughness exponents for the two substrates are almost the same within the error. The data of maximal height distributions (minmal height distribution) on the width of the saturated surface for the two fractal substrates can be well collapsed together and fitted by Asym2Sig distribution.