the imperial templars are just soliders they are not like the southern templars.
Are they just soldiers?
I get that they are work by different rules, enforcing the law on behalf of the Magisterium but not supposed to act on their own, and that they do not have the full powers against magic that the Orlesian Chantry templars do, as made explicit in Inquisition if you choose to send covert templars to aid a Tevinter magister against his opponent.
So one might think of them as policemen of some sort.
On the other hand, they are the ones tasked with acting against rogue mages in the empire and can enforce the Right of Annulment according to Fenris in Dragon Age:2, which doesn't sound like the sort of job to hand to ordinary policemen or soldiers, and Dorian in Inquisition says how the imperial templars are mostly used against mages that fall out of favour with the magisterium or are weak (because nobody dares touch those that are strong and in favour) rather than based on the magnitude of their crimes.
All taken together, though it is made explicit that they do not have the full powers against mages that the southern Templars do, what I've been told in-game in the last two games just doesn't square with Tevinter templars being ordinary soldiers, as differences aside, they are a weapon used by mages against other mages.