We have combined observations of the Antennae galaxies from the radio interferometer ALMA (Atacama Large Millimeter/submillimeter Array) and from the optical interferometer GHαFaS (Galaxy Hα Fabry-Perot System). The two sets of observations have comparable angular and spectral resolutions, enabling us to identify 142 giant molecular clouds (GMCs) and 303 H II regions. We have measured, and compared, their basic physical properties (radius, velocity dispersion, luminosity). For the H II regions, we find two physical regimes, one for masses >105.4 M⊙ of ionized gas, where the gas density increases with gas mass, the other for masses <105.4 M⊙ of ionized gas, where the gas density decreases with gas mass. For the GMCs, we find, in contrast to previous studies in other galaxies over a generally lower mass range of clouds, that the gas density increases with the total gas mass, hinting at two regimes for these clouds if we consider both sources of data. We also find that the GMC mass function has a break at 106.7 M⊙. Using the velocity dispersion measurements, we claim that the difference between the regimes is the nature of the dominant binding force. For the regions in the lower mass range, the dominant force is the external pressure, while in the higher mass range it is the internal gravity of the clouds. In the regime where gravity is dominant, the star formation rate, derived from the dust-corrected Hα luminosity, increases super-linearly with the velocity dispersion, and the gas density increases with the gas mass.