Recent experiments on polyelectrolyte membranes have clearly shown that at operating temperatures perfluoroimide acid (PFIA) has a higher electrical conductivity than widely used Nafion. In the present paper classical molecular-dynamics simulations were carried out to study the structural properties of both materials, and the proton and water transport in the corresponding membranes at T=300 K and T=353 K. In this temperature range, the temperature effects on the hydrated internal polyelectrolyte structure were found to be negligible. The PFIA has longer side chains across a wide range of hydration levels which would have promoted more trapping of water and hydronium ions in PFIA. Indeed, the average number of water molecules in the first hydration shell around the side-chain protogenic groups was found to be higher in PFIA than in Nafion. Our simulations showed the formation of large continuous water clusters and connected pore volumes in PFIA at high hydration levels which promotes conductivity. The diffusivity constants for hydronium ions and water increase with increasing hydration and increasing temperature. Unlike the experimental conductivities, the simulated data for PFIA were comparable to those of Nafion at high hydration levels. Note that the experimentally measured conductivity in PFIA is both due to vehicular transport of ions, which can be resolved using classical molecular dynamics, and structural Grotthuss diffusion, which cannot be resolved in our simulations. Interestingly, we observed a higher total number of water molecules in the first coordination shell around hydronium in PFIA than in Nafion at higher hydration levels. This should aid in more hydrogen bonding between hydronium and water in PFIA which, in turn, should help in structural diffusion. Finally, we discuss our preliminary results and some peculiarities of the proton transport in Nafion membranes filled with the graphene oxide nanoflakes.