These are largely just different terms for the same things, but named differently to remind you of some important details. For a rectangle, we usually refer to the length and width; properly, these are the longer side and the shorter side, respectively, but that really makes no difference.
John Adams hated political parties, but he was backed by the Federalists and ran for President as a Federalist and won.   After Adams served, Thomas Jefferson, James Madison James Monroe served two terms each as members of the Democratic Republican party. By this time the Federalist Party all but disappeared.
President Benjamin Harrison 23rd President. Benjamin Harrison, a member of the Republican Party, took office as the 23rd President of the United States on March 4, 1889 at age 56. Harrison served in office for 4 years and left when he lost reelection. He was born in North Bend, Ohio and received an education from Miami University of Ohio.