cell phone is American English. Mobile phone is British English.

As a follow-up question: What do Australians, New Zealanders, Canadians, Nigerians, South Africans, West Indians, and other English speakers say? Mobile phone or cell phone? I would guess that Canadians say cell phone and New Zealanders say mobile phone. And I bet Nigerians say mobile phone too. But I'm not sure about the others.

They are the same thing. Mobile phone calls are transferred from one transmitter to another as you travel. Each transmitter has a range of a certain number of miles. That area is called a cell. As you travel your phone is MOBILE and the call is transferred from CELL to CELL. The British focus on the MOBILE aspect, the Americans focus on the CELLULAR nature of the system. Hence there are two names for the same thing.