We report quantum dynamics calculations of rotational and vibrational energy transfer in collisions between two para-H(2) molecules over collision energies spanning from the ultracold limit to thermal energies. Results obtained using a recent full-dimensional H(2)-H(2) potential energy surface (PES) developed by Hinde [J. Chem. Phys. 128, 154308 (2008)] are compared with those derived from the Boothroyd, Martin, Keogh, and Peterson (BMKP) PES [J. Chem. Phys. 116, 666 (2002)]. For vibrational relaxation of H(2)(v=1,j=0) by collisions with H(2)(v=0,j=0) as well as rotational excitations in collisions between ground state H(2) molecules, the PES of Hinde is found to yield results in better agreement with available experimental data. A highly efficient near-resonant energy transfer mechanism that conserves internal rotational angular momentum and was identified in our previous study of the H(2)-H(2) system [Phys. Rev. A 77, 030704(R) (2008)] using the BMKP PES is also found to be reproduced by the Hinde PES, demonstrating that the process is largely insensitive to the details of the PES. In the absence of the near-resonance mechanism, vibrational relaxation is driven by the anisotropy of the potential energy surface. Based on a comparison of results obtained using the Hinde and BMKP PESs with available experimental data, it appears that the Hinde PES provides a more accurate description of rotational and vibrational transitions in H(2)-H(2) collisions, at least for vibrational quantum numbers v ≤ 1.