The similarity between vegetative and brain-dead patients lies in the fact that both are unresponsive to the external environment.
The major difference between the two, however, is that the former still retains some hypothalamic and brainstem function, has spontaneous breathing and sleep and wake cycles, and recent research suggests that a small number of vegetative patients may even be conscious, and that these residual functions may allow them to survive for a long time with adequate medical care.
Brain death, on the other hand, is the loss of all brain function, followed by short-term failure of other organs and eventual cardiac arrest.