They didn't, as a whole. The earliest version of their Rule—the document that contains the laws dictating its membership—, dating from a mid thirteenth-century manuscript, contains no provisions regarding what armor its members ought to wear, and later revisions did not add any section on the matter. Here is a translation of the Middle High German from a Indrikis Stern's 1969 dissertation on the topic. Thus members of the Teutonic Order likely wore the medieval helmets of all types. Sven Ekdahl has argued forcefully that the Order adapted itself to local conditions in Prussia, breeding the local variety of horse and using conical-style helmet favored by the local Prussians. All of which suggests that helmet choice was reserved to the personal and the practical rather than a uniform.
During the high middle ages, knights got a little crazy with their helms from time to time. Tournaments in particular were times when knights sought to show off both their martial prowess and their status, and they often tried to combine the two with increasingly elaborate tournament helms. Looking through the early fourteenth-century Codex Manesse, we can get a glimpse of the scope of this madness. For example, Ulrich von Lichtenstein, a knight and poet, rode onto the tournament field on a stallion draped in green whilst wearing a pink Mermaid. Other competitors wore dragons, hawks, and Wachsmut von Künzingen had the gall to mount what appears to be two sets of fish on his helmet. The manuscript includes a Teutonic knight known as Tannhäuser, who can be seen in the habit of his order with a picture of his winged helmet, which marked his lineage from Swabian nobility—although, to be completely frank, there is too much disagreement and guesswork around what family exactly that is. Fun as these helmets were, it should be emphasized that they were not likely worn into battle, but are best understood as subset of heraldry, a sort of ephemeral "fad" in late thirteenth century tournament fashion and pageantry.
It is likely in this spirit of heraldric display and personal bravado that Albert von Pranckh stuck great buffalo-style horns onto his otherwise ordinary bucket helmet. The horns were a part of the Pranckh heraldry that continue to be a part of the family coat of arms even today, with some additions. Albert likely used it in much the same way we see individuals using their fancy helmets in the Codex Manesse, at tournament or other public events of knightly prowess. After his death, the helmet was used by a cadet branch of the Pranckh family as part of their burial ritual, hence why its often referred to as a "funerary helmet."
What I can't figure out is how the design of Pranckh's helmet became so attached to the contemporary images of the Teutonic Order, since Albert wasn't himself a member of the order as far as I can tell. I suspect the answer comes from early depictions of Teutonic knights in film. Sergei Eisenstein directed Alexander Nevsky in 1938, and—although I haven't seen it—images of the film feature costumes of Teutonic knights wearing ridiculous helmets. It is likely that when researching for the film Eisenstein found inspiration in Albert's helmet.
For the early history of the Order, I would look no further than Nicholas Morton's The Teutonic Knights in the Holy Land, 1190-1291 (2009). There is also Sven Ekdahl's chapter on the military history of the Order in Prussia. Its pretty dry stuff in my opinion, but he has some fascinating little details concerning equipment. See: Sven Ekdahl, "Horses and Crossbows: Two Important Warfare Advantages of the Teutonic Order in Prussia" in The Military Orders Volume II, 119-152
