Answer:
The Prime Minister is the head of the Executive branch of the Government in the United Kingdom as well as other Parliamentary or Semi-presidential Government systems.
The President is the head of the Executive branch of the Government in the United States and other Republican states.
The differences between the two especially between the United Kingdom and the United States is;
The President is elected directly by the will of the people which in the United States is represented by the Electoral College. The Prime Minister on the other hand is elected if their party and affiliates hold a majority in the House of Commons (Parliament).
The Presidential election in the United States is held every 4 years with the next one being this year, 2020. By contrast Parliamentary elections in the UK are held every 5 years but this can be less if the Prime Minister dissolves parliament.
As the Prime Minister is elected by their party's representatives in Parliament, they can also be removed by them. This happens when the members pass a vote of no confidence in the the minister.