Recent synthesis of the endohedral complexes of C-70 and its open-cage derivative with one and two H-2 molecules has opened the path for experimental and theoretical investigations of the unique dynamic, spectroscopic, and other properties of systems with multiple hydrogen molecules confined inside a nanoscale cavity. Here we report a rigorous theoretical study of the dynamics of the coupled translational and rotational motions of H-2 molecules in C-70 and C-60, which are highly quantum mechanical. Diffusion Monte Carlo (DMC) calculations were performed for up to three para-H-2 (p-H-2) molecules encapsulated in C-70 and for one and two p-H-2 molecules inside C-60. These calculations provide a quantitative description of the ground-state properties, energetics, and the translation rotation (T-R) zero-point energies (ZPEs) of the nanoconfined p-H-2 molecules and of the spatial distribution of two p-H-2 molecules in the cavity of C-70. The energy of the global minimum on the intermolecular potential energy surface (PES) is negative for one and two H-2 molecules in C-70 but has a high positive value when the third H-2 is added, implying that at most two H-2 molecules can be stabilized inside C-70. By the same criterion, in the case of C-60, only the endohedral complex with one H-2 molecule is energetically stable. Our results are consistent with the fact that recently both (H-2)(n)@C-70 (n = 1, 2) and H-2@C-60 were prepared, but not (H-2)(3)@C-70 or (H-2)(2)@C-60. The ZPE of the coupled T-R motions, from the DMC calculations, grows rapidly with the number of caged p-H-2 molecules and is a significant fraction of the well depth of the intermolecular PES, 11% in the case of p-H-2@C-70 and 52% for (p-H-2)(2)@C-70. Consequently, the T-R ZPE represents a major component of the energetics of the encapsulated H-2 molecules. The inclusion of the ZPE nearly doubles the energy by which (p-H-2)(3)@C-70 is destabilized and increases by 66% the energetic destabilization of (p-H-2)(2)@C-60. For these reasons, the T-R ZPE has to be calculated accurately and taken into account for reliable theoretical predictions regarding the stability of the endohedral fullerene complexes with hydrogen molecules and their maximum H-2 content.