The Teutonic Order was established as a hospital for German pilgrims and soldiers after the aborted German crusade during the 3rd Crusade. Soon after it took on a military function.

The Teutonic Order's proper name is The Order of the Hospital of St. Mary of the Germans in Jerusalem. When the Order Militarized it modeled itself on the Templars.

Functionally, the Orders have more in common than different but their histories are very different

Source: The Teutonic Knights: A Military History by William Urban