Molecular-dynamics simulation results on thermodynamic and transport properties of pure H2S under conditions of practical interest are presented. Our data are in very good quantitative agreement with the scarce experimental data and estimates on thermophysical properties of this substance. Our results serve as a test of the validity of the intermolecular potential used in the simulations as well as the consistency of the existing data in the studied range. New simulation data on thermal conductivity at low temperature as well as in supercritical states are also reported. Furthermore, we present a comparative analysis between the local order in the liquid phase of pure hydrogen sulfide and water, due to the molecular analogies between both substances, and its relation with the formation of H=S bonds. Our results indicate that under the same corresponding thermodynamic states, H2S is a much less structured substance, with a first solvation shell with a dodecahedral order instead of the tetrahedral order observed in water.