We present here a study of the diffusive motion of a particle submerged in bistable and metastable periodic potentials within the framework of the Brownian motion theory. This is done through an investigation of the quasi-elastic peak in the dynamic structure factor S-S(q,omega). Its width is found to contain valuable information on the mechanism for the diffusion process. For this study, we use the Fokker-Planck equation, which is solved numerically by the matrix-continued fraction method (MCFM) for wide system parameters' range and for the both types of potentials. It is the purpose of the present work to study how the transport properties of the system are modified by going from bistable to metastable periodic potential. Our finding results indicate large difference between transport properties in bistable and metastable potentials essentially at low temperature. In fact, in the former case, the mechanism process results from a combination of inter-cell liquid-like and intra-cells hopping motion of the particle. While for the second case, the diffusive process consists only of hopping motion, with different jump lengths, inside and between the cells. So, in metastable potential, a simple jump model describes the diffusive motion quite well. Further, a direct comparison between the numerical diffusion coefficient D and the analytical approximation for both potential shapes in the intermediate friction limit is presented and discussed.