The hydrogen molecule dynamics in tetrahydrofuran-H2-H2O clathrate hydrate has been studied by high-resolution inelastic neutron scattering and Raman light scattering. Several intense bands in the neutron spectrum are observed that are due to H2 molecule excitations. These are rotational transitions, center-of-mass translational transitions (rattling) of either para- or ortho-H2, and combinations of rotations and center-of-mass transitions. The rattling of the H2 molecule is a paradigmatic example of the motion of a quantum particle in a non-harmonic three-dimensional potential well. Both the H2 rotational transition and the fundamental of the rattling transition split into triplets. Raman spectra show a similar splitting of the S0(0) rotational transition, due to a significant anisotropy of the potential with respect to the orientation of the molecule in the cage. The comparison of our experimental values for the transition frequencies to a recent quantum mechanical calculation gives qualitative agreement, but shows some significant difference.