We have performed a rigorous theoretical study of the quantum translation-rotation (T-R) dynamics of one and two H-2 and D-2 molecules confined inside the large hexakaidecahedral (5(12)6(4)) cage of the sII clathrate hydrate. For a single encapsulated H-2 and D-2 molecule, accurate quantum five-dimensional calculations of the T-R energy levels and wave functions are performed that include explicitly, as fully coupled, all three translational and the two rotational degrees of freedom of the hydrogen molecule, while the cage is taken to be rigid. In addition, the ground-state properties, energetics, and spatial distribution of one and two p-H-2 and o-D-2 molecules in the large cage are calculated rigorously using the diffusion Monte Carlo method. These calculations reveal that the low-energy T-R dynamics of hydrogen molecules in the large cage are qualitatively different from that inside the small cage, studied by us recently. This is caused by the following: (i) The large cage has a cavity whose diameter is about twice that of the small cage for the hydrogen molecule. (ii) In the small cage, the potential energy surface (PES) for H-2 is essentially flat in the central region, while in the large cage the PES has a prominent maximum at the cage center, whose height exceeds the T-R zero-point energy of H-2/D-2. As a result, the guest molecule is excluded from the central part of the large cage, its wave function localized around the off-center global minimum. Peculiar quantum dynamics of the hydrogen molecule squeezed between the central maximum and the cage wall manifests in the excited T-R states whose energies and wave functions differ greatly from those for the small cage. Moreover, they are sensitive to the variations in the hydrogen-bonding topology, which modulate the corrugation of the cage wall.