Quantum close-coupling and coupled-states approximation scattering calculations of rotational energy transfer in CO due to collisions with H2 are presented for collision energies between 10(-6) and 15,000 cm(-1) using the H2-CO interaction potentials of Jankowski and Szalewicz [J. Chem. Phys. 123, 104301 (2005); 108, 3554 (1998)]. State-to-state cross sections and rate coefficients are reported for the quenching of CO initially in rotational levels j2 = 1-3 by collisions with both para- and ortho-H2. Comparison with the available theoretical and experimental results shows good agreement, but some discrepancies with previous calculations using the earlier potential remain. Interestingly, elastic and inelastic cross sections for the quenching of CO (j2 = 1) by para-H2 reveal significant differences at low collision energies. The differences in the well depths of the van der Waals interactions of the two potential surfaces lead to different resonance structures in the cross sections. In particular, the presence of a near-zero-energy resonance for the earlier potential which has a deeper van der Waals well yields elastic and inelastic cross sections that are about a factor of 5 larger than that for the newer potential at collision energies lower than 10(-3) cm(-1).