The difference is simple, and not very subtle; the Templars ceased to exist, and the Hospitallers certainly exist right now, working for the sick. Originally the latter were of a military order, the Knights Hospitallers of St. John of Jerusalem. The name comes from the dedication to St. John the Baptist of their headquarters in Jerusalem.
These are not their only names: from 1310 they were the Knights of Rhodes, and from 1530 the Knights of Malta, but they were established themselves first in (or around) 1070 with Muslim permission, managing a hospital for sick pilgrims in Jerusalem. They only became a formal order of knights when the city fell to the first Crusaders in 1099.
They wore a black habit, with a white eight-pointed Maltese Cross. They elected a Master and under him were at first purely military, in an order which spread quickly across Europe. In questions of order and discipline they followed Augustinian rules (q.v.) and divided themselves into three classes or ranks: knights, chaplains and serving brothers.
Driven out of Jerusalem by Saladin himself they moved to Acre, from which they were expelled a century later, transferring to Cyprus. In 1310, however, they captured the island of Rhodes and remained there until 1522. Then Emperor Charles V made them a present of the island of Malta, which they had to defend by force against the Turks, but they could not deal in a similar fashion with Napoleon: by this time the Order had lost its influence and supporting voices.
Some members moved to Russia where they made Paul I Grand Master of their Order. He died in 1801 and the order collapsed temporarily in confusion, only to be restored again in England in the 1830s. Only the English branch now remains, and the aim of the Order, in the persons of doctors, surgeons, nurses etc., is to care for the sick. They stopped their military function around the end of the eighteenth century.
The case of the Knights Templar is altogether different. The Order was founded around 1118 by Hugh de Payens or Payns, a knight from the region of Champagne in France. He called it The Poor Knights of Christ and the Temple of Solomon; he and seven companions promised to protect pilgrims travelling on the public roads in The Holy Land. In 1128 official approval was provided by the Treaty of Troyes on the understanding they they would follow Benedictine rules (q.v.) Here is the first notable difference: the Hospitallers were Augustinian, and Templars Benedictine.
The Templars rose rapidly in popularity, attracting many noble new members; the Order became very rich, acquiring much property throughout the Christian sphere. After the fall of Jerusalem in 1387 they moved to Acre with the Hospitallers – a grave mistake. Rivalry and mutual loathing developed between the two Orders, not without violent encounters. When the Hospitallers moved to Cyprus, so did the Templars, which was also a seriously bad career move. On this island they became bankers rather more than soldiers, and attracted half the nobility of Europe to invest with them. The Hospitallers complained that what was supposed to be a religious order was now simply making a fortune and had ceased to be Christian or even religious in tone.
Sadly, they attracted the attention and disapproval of a French king, Philip IV of France (q.v.), was decidedly hostile, for reasons that can only have been financial. Clever as ever,
Philip ‘the Fair’ persuaded Pope Clement to suppress the Order, enabling the wily Frenchman to grab their enormous wealth and property. One assumes that the Pope insisted on sharing it with ‘The Fair One’.
The Templars were assaulted in their castles, mostly killed, and their Grand Master was burnt at the stake for heresy. Historians of the epoch wrote of them as blasphemers, practitioners of Black Magic, sexually depraved (historians should know), cruel and unjust. Not surprisingly, given a very bad press, they failed to form up again and vanished from History, though the Ku Klux Klan in the US claimed to follow their example and creed. Such nonsense.