I am currently enrolled in a MAT History program and I am kind of lost when it comes to the differences in jobs that Historians and teachers do. It feels like the most obvious one might be that Historians work closely with the archive and teachers don't always, though I believe in incorporating more primary sources in a classroom. But another distinction could be that Historians create a text and all teachers do is teach. But even then teachers can still create a narrative for their classrooms and that is dependent on which texts or sources they are drawing from.
So I'm wondering: Is there a difference and why or why not?
Historians specifically do primary source research, as you say. Basically, they're researchers of history. Their main goal is to research historical questions, trying to add new sources, new perspectives and new analysis to the existing body of knowledge.They write professional level books and articles about historical subjects, attend conferences to talk about historical questions, and they're almost always university professors and so they teach too. Historians work within universities, teach university students, and concentrate heavily on research.
History teachers, on the other hand, will usually work in high schools or elementary schools. They don't research to discover new info, they research info for their courses then teach, and they use materials that come from the historians' books and research and primary source collections. Teachers generally also know a lot about history, but they're less specialized than historians; a teacher, for example, would have to know about the general overview of history to be able to do the curriculum the government assigns and answer foreseeable questions, but a historian will concentrate on a very specific little part of history and endeavour to know as much as possible about it, then write books to share that knowledge with others. Because of this, their training differs: a historian will get a PhD in a very specific subsection of history, whereas a history teacher will have taken a certain number of history class at the undergrad level and then gotten a Bachelor of Education to know good teaching techniques.
But a lot of historians are also really good/dedicated at teaching in the university classes they offer! So there's some overlap in the sense that they both do a bunch of teaching. Different grade levels, though.
Your confusion is probably a result of most historians also being history teachers in some capacity. History is about interpreting knowledge from the past and then disseminating that information in the present. In this way, both historians and history teachers are 'doing history'. The former produces work primarily for a highly-educated, expert audience (including history teachers), where the latter produces work primarily for a novice audience.
The distinction in your case, however, probably has to do with degree requirements more than anything else. Many MAT programs require students to complete a substantial teaching portfolio or other major pedagogical project, whereas MA programs typically require a major, original writing assignment, such as a thesis, that demonstrates expertise in the field. Both routes often require comprehensive exams to show content competency before these major projects.
Furthermore, for a vast majority MA students, since the minimum requirement for professional historians is often a PhD., they have a few more years to receive additional training and further refine their research and writing skills before going on the job market. An MAT History degree makes you immediately marketable to teach history at a variety of levels (depending on certification).