A detailed comparison between three-dimensional classical surface hopping calculations and quantum mechanical calculations is presented for the photodissociation of water in the (B) over tilde band. Accurate coupled diabatic potential energy surfaces a-re used in these calculations. Tully's "fewest switches" method using an adiabatic representation for the electronic states is applied for the surface hopping procedure. Studied are the energy dependence of the branching ratios for the possible fragmentation channels, including electronically nonadiabatic channels, and the probabilities for particular vibrational or rotational product states of the electronically excited OH(A) fragment. Although the classical results generally agree well with the quantum results, some serious errors in the classical calculations were detected. First, it is found that the calculated fractions for the O(D-1) + H-2 and O(P-3) + H + H fragments are too large. Second, the absence of quantization of the vibrational energy in classical mechanics has consequences for the details of the rotational product state distribution of the OH(A,nu=0) fragments. This is important for the "single N phenomenon", an experimentally observed strong preference for populating the highest rotational product state for which the rotational barrier energy is lower than the available energy (S. A. Harich, X. F. Yang, R, van Harrevelt, and M. C. van Hemert, Phys. Rev Lett., 2001). For a two-dimensional model, where the above-mentioned problems of classical trajectory calculations do not occur, excellent agreement between classical and quantum results, is found, Classical trajectories were followed to explain the single N phenomenon and the origin of the experimentally observed vibrational excitation of OH(A) fragments.