A man with the same name as his father uses “Jr.” after his name as long as his father is alive. His father may use the suffix “Sr.” for “senior.” The son may either drop the suffix after his father’s death or, if he prefers, retain it so that he won’t be confused with his late father." - Emily Post Etiquette

A "Jr." is only a "Jr." while the father is alive. After the father dies, the "Jr." drops off and the son is known only by the name.