The first Crusade started in 1096 and lasted until 1099, the causes of this are widely debated but I was taught that the Crusades were launched in response to the Seljuk victory at Manzikert which saw a large portion of Anatolia(Modern day Turkey) fall into Muslim hands. When the Emperor of the Byzantime Empire asked the Pope for help, the Pope thought this would usher in a reunification of the Catholic and Eastern Orthodoxy church, so he called for a Crusade to defeat the Muslims, and reclaim the holy city of Jersualem, which had been under Muslim control for over 400 years at this point in time. Ultimately the reunification between the Catholic and Eastern Orthodoxy never ocurred, and even Constantinople was sacked by a Crusading army in the future, but your question is about two of the many military orders that rose up in this time.

In, 1119, the French knight Hugues de Payens petitioned King Baldwin of Jerusalem with the notion of creating a holy militant order to protect pilgrims making the journey from Europe. Though the city of Jersualem, Acre, and Antioch in the north all were held by Christian's and upstart Crusading Kingdoms, the lands were stricken with a bandit plague. These bandits would often make short work of peasants and other Christian Pilgrims coming in from Europe. So in order to combat this the Knights Templar were created.

The Templars were renowned warriors, feared by Muslims and praised by Christians everywhere. They grew in such power that they rivaled the Kings in Europe. Chapter houses were opened throughout the Crusader States as well as in Europe. They had a strict code that men of the order would own no castles, hold no lands, and never marry. So the order was made up primarily of second sons who wouldn't have inherited their fathers lands anyway, so striking out to make a name for themselves, they pledged their undying loyalty to God and to the Order. They opened Hospitals and they cared for Christian pilgrims everywhere. As good as they might seem, they were also brutal towards their enemies, but even they would put aside their loyalties to help the Muslims combat the later Mongol invasion of the middle east.

Many other Knightly Orders rose in this time, including the Teutonic Knights who were not known for what they did in the Holy Land, but rather what they did in Lithuania and Poland.

The Teutonic Order was created in 1192, in Acre, comprised mostly of German Knights. They mostly helped build hospitals to house the German pilgrims coming to the Holy Land. Once the Holy Land was lost to the Saracens, the Teutonic Order retreated to Transylvannia, and then eventually to Northern Germany and modern day Poland. There they guarded German missionaries from Prussian pagans who would often raid and kill all Priests and Monks who tried to convert them. Overtime the Teutonic Order eventually came to start recruiting more and more Prussian converts into their ranks and began to build up more and more power. Again copying what the Templars did, the Teutons built chapter houses all over Germany, but their true house of power was Königsberg (Kaliningrad, Russia).

Eventually the Teutonic Knights began to become more and more power hungry and began to brutalize their new enemy, being the Lithuanian pagans. They even began to move against the Polish Kingdoms, which were Catholic, and the Novgorodian Princes who were Eastern Orthodox. Eventually the Polish and Lithuanians defeated the Knights at the Battle of Grunwald which cost them much of their power, until their Livonian Branch was defeated by Alexander Nevesky almost 100 years after Grunwald which finally ended the Orders claims over Russian lands in the North.

Fundamentally The Teutonic Order was based on the Templars with each member refusing land and wealth all in the service of God. The main difference between them was the fact that the Templars never recruited "auxiliaries" not even Turccopoles, to fight against the swifter horse archers of the Muslim and Mongol armies. While the Teutonic Order heavily relied on auxiliary troops.

Ultimately both ended up being too powerful for their own good and created many enemies by being so. The difference was that the Teutonic Order was much more aggressive with its neighbors, while the Chapter houses of the Templar order were mainly just that, never really lands to rule just lands to recuperate and launch another crusade against the Holy Land.

Another big difference between them is money and the banking system that the Templar order built. This banking system gave the Templar a lot more influence over the rest of Europe and much more organized that the Teutonic Order could ever have hoped to have been. The Teutonic Knights also waged war on the Catholic Kingdom of Poland over territorial disputes, which the Templar Order most likely would have just used gold to solve the issue rather than annihilate Christians. That however is just a guess and its not based upon fact, it very well could have been that the Templar order may have found itself in the exact position as the Teutonic Knights found themselves. However I do not believe that the Teutonic Order would have found the same end as the Templars due to their order being much more disorganized and not as well financed.