Here we study and compare nonlocal diffusion processes on networks based on two different kinds of Laplacian operators. We prove that a nonlocal diffusion process on a network based on the path Laplacian operator always converges faster than the standard diffusion. The nonlocal diffusion based on the fractional powers of the graph Laplacian frequently converges slower than the local process. Additionally, the path-based diffusion always displays smaller average commute time and better diffusive efficiency than the local diffusive process. On the contrary, the fractional diffusion frequently has longer commute times and worse diffusive efficiency than the standard diffusion process. Another difference between the two processes is related to the way in which they operate the diffusion through the nodes and edges of the graph. The fractional diffusion occurs in a backtracking way, which may left the diffusive particle trapped just behind obstacles in the nodes of the graph, such as a weighted self-loop. The path-diffusion operates in a non-backtracking way, which may represent through-space jumps that avoids such obstacles. We show that the fractional Laplacian cannot differentiate between three classes of brain cellular tissues corresponding to healthy, inflamed and glioma samples. The path Laplacian diffusive distance correctly classifies 100% of the mentioned samples. These results illuminates about the potential areas of applications of both kinds of nonlocal operators on networks.