Rotational excitation of the interstellar species SO(X3Sigma-) with H2 is investigated. The authors present a new four-dimensional potential energy surface for the SO-H2 system, calculated at an internuclear SO distance frozen at its experimental minimum energy distance. It was obtained at the RCCSD(T) level using the aug-cc-pVTZ basis sets for the four atoms. Bond functions were placed at mid-distance between the SO center of mass and the center of mass of H2 for a better description of the van der Waals interaction. Close coupling calculations of the collisional excitation cross sections between the fine structure levels of SO by collisions with para-H2 are calculated at low energies which yield, after Boltzmann thermal average, rate coefficients up to 50 K. The exact level splitting is taken into account. The propensity rules between fine structure levels are studied. It is shown that F-conserving cross sections are much larger, especially for high-N rotational levels, than F-changing cross sections, as found previously for SO-He collisions and expected from theoretical considerations. The new rate coefficients are compared with previous results obtained for this molecule and they find that important differences exist that can induce important consequences on astrophysical modeling. Comparison with excitation by collision with He shows that the rate coefficients differ by important factors that cannot be only explained by the reduced mass ratio in the thermal average. This may be due to differences between the potential energy surfaces as well as to the contribution of the different reduced masses in the scattering equations.