Despite the Hospitallers' headstart, the Templars soon eclipsed them. This was probably the origin for the later antipathy between the two orders. Usually, their rivalry was only political, but sometimes it could lead to civil war.
In Palestinian politics, the two orders were often found on opposite sides. One of the most significant periods of conflict occurred during the reign of Baldwin IV of Jerusalem, the leper King. Upon Baldwin's death in 1185, the Templars favored the succession of Baldwin's sister, Sybil, and Guy de Lusignan, her husband. The Hospitallers supported the other side, which desired the election of a new ruler from among the Palestinian barons. (...)Increasingly, the two Orders each made their own treaties with the Muslims, which often contradicted the interests of the remaining local nobility and of the other religious orders. This rivalry was occasionally alleviated by comradeship and loyalty between individuals, even to the detriment of each Order's internal policies. Occasionally, they even went into battle against the Muslims together.
The two Orders were united in one other thing. In the 13th century, they were the largest land owners in Palestine. The Templars, alone, held nearly a third of the Frankish land in the region. As the Palestinian barons had either gone bankrupt, or died out, the military orders had been forced to buy their land, to keep it out of the hands of the Muslims. Some critics accused them of deliberately impoverishing local Franks in order to acquire more territory. This seems unlikely. The orders complained often about the strain this extra burden put on their finances and manpower.
This blog presents slightly edited and/or rearranged quotes from the pp 43-44 of Stiles, Paula Regina, "BETWEEN TWO FAITHS: THE ARABIZATION OF THE KNIGHTS TEMPLAR DURING THE CRUSADES" (1999). Open Access Master's Theses. Paper 1805. The illustration shows a Templar and a Hospitaller Knight source