To sum up, lift and elevator can be used as synonyms. The major difference between these two is its linguistic usage, i.e. lift is used as a British term, and the elevator is used as an American term.
elevator, also called lift, car that moves in a vertical shaft to carry passengers or freight between the levels of a multistory building. Most modern elevators are propelled by electric motors, with the aid of a counterweight, through a system of cables and sheaves
