Methane hydrate was recently shown, both experimentally and through simulations, to be stable up to the remarkably high pressure of 150 GPa. A new methane hydrate high-pressure (MH-IV) phase, reminiscent of ice at ambient pressure, was described for pressures above approximately 40 GPa. We disentangle here the main contributions to the relative stability of the lower pressure, denoted MH-III and the high-pressure MH-IV structures. Through several simulation techniques, including metadynamics and Path Integral Molecular Dynamics for nuclear quantum effects, we analyze the phase transition mechanism, which implies hydrogen bond breaking and reforming, as well as methane reordering. The transition pathway is far from trivial and the quantum delocalization of the hydrogen nuclei plays a significant role.