The quantum and the classical multiphoton dissociation dynamics of diatomic Morse molecules driven by intense and ultrashort two-color laser pulses are investigated and compared. Special attention is given to the role of the relative phase of the monochromatic components of the pulses. We study the excitation of the system from the vibrational ground-state and from excited states for several values of amplitude, frequency and duration of the external pulses. Similar overall sensitivity of the dissociation threshold on the phase is observed in both quantum and classical approaches, provided the excitation frequency is sufficiently far from quantum resonances. In addition, we analyze the correspondence between the effects of the relative phase on the classical and quantum dynamics.