In 1916, President Woodrow Wilson established June 14th as Flag Day. That was later confirmed in 1949 by the US Congress. It is not an officially recognized Federal Holiday though.
The flag depicted above is the flag flying over the USS Niagara and it is the configuration of the US Flag at the time of the War of 1812.
On June 14th, 1775, the Continental Congress establishes the Continental Army. This day is officially recognized as the birthday of the United States Army. After the conclusion of the Revolutionary War, the US Congress passes the act that creates the United States Army on June 4, 1784. Because the US Army considers its roots as the Continental Army, it considers the earlier date their official birthday.