During depletion-driven vesicle adhesion, a stiff membrane's resistance to bending at fixed tension prevents contact angle equilibrium and vesicle spreading over an opposing vesicle, while more flexible vesicles partially engulf opposing vesicles. Estimates of the bending cost associated with the spreading contact line, relative to the adhesion energy, were consistent with the observed spreading or lack of spreading for the flexible and stiff membranes, respectively, and predicted a lag time sometimes preceding spreading.