Honestly, this is a silly argument. On a one-on-one scale like this, you are basically asking who would win between a fiercely religious and well-trained German knight and a fiercely religious and well-trained French knight. At the individual level, the only significant difference is their average nationalities. They would both wear state-of-the-art armour, be trained to the highest degree, and they were both warrior monks (with rules that are highly intertwined), so they would have the same level of religious commitment. The winner is going to be whoever is on their game in that day, or gets lucky, or God knows what else happens. It's pointless.
And to answer the basic questions I've seen here, the Teutonic Knights were founded at the siege of Acre in 1191, with a military function based on Templars, as a German branch of the Hospitallers. They became their own Order shortly afterwards, and led the crusades against pagans. They were subservient to both the Pope and the Emperor, but in practice they were highly autonomous.
On a larger scale you can compare the Templars and Teutonic Knights and actually make some conclusions there. For instance, two interesting facts are that the Templars actually have a loosing record for the large battles they have participated in, and the Teutonic Knights led the only successful crusade (all the while participating in the loosing crusade, which was ran by a certain different order). Read from that what you will.
