@Richman They are technically distinct:
Living corpse - The body is biologically alive, but the personality is not. Could range from a poetic description of deep depression to actual brain death with the body kept alive through mechanical means. Alive, but inert.
Living dead - The body is biologically dead but has been reanimated through magical, scientific, or unexplained means. Sometimes this is used in a metaphorical sense to refer to people who are shambling their way through life. Dead (or "dead") but active.