..."guilds"...? Guilds are, you know, craft guilds. As in craftsmen. As in armourers, tanners, and whatever gazillion other professions medieval urban centers had that were influential and assertive enough to form a separate corporative organization (and not be part of a bigger guild).

The word is "Orders". As in knightly.

Anyway, I don't think the Templars and Teutonics had too much friction overall. Their spheres of interest were just too diverse for there to have been the sort of constant tension and squabbling that went on between the Templars and the Hospitallers (aka Johannites) who both had major territories in Outremer and constant disputes of precedence, prestige and practical political and military issues.

Both also tended to not quite see eye to eye with various crown-heads - and sometimes ecclesiarchical authorities - over various matters.

The Teutonic Order mostly concerned itself with the remaining pagans of northern Central Europe - the Baltic coastline and Lithuania - whereas the Templars and Hospitallers primarily busied themselves with fighting Muslims in the Mediterranean theatre, so they didn't butt heads all that often.