Sometimes one bumps into the name "geometric algebra" (henceforth "GA"), in the sense of this wikipedia article. Other names appear in that context such as "vector manifold", "pseudoscalar", and "space-time".
A very superficial look at that wikipedia article, or books on the topic, gives me the idea that it is essentially about Clifford algebras and related calculus. One impression that I got (but I could be wrong) is that there is a relatively small group of authors (I don't know if mathematicians or physicists or both) that have produced work in "GA" and that this group is probably disjoint from the set of mathematicians who wrote about the algebraic foundations of Clifford algebra theory or about Clifford algebras featuring in contexts such as the Atiyah-Singer index theorem or Clifford analysis (the study of Dirac-type operators). Also, I'm not aware if any reference to the field "GA", as such, appears outside works specifically designated as "GA" and written by people in that group. It's also not clear to me if there is an intersection (or even subset relation) between "GA" and the above mentioned areas of mathematics and the extent of such intersection.
1. Is there anything in the field "geometric algebra" that is distinct from usual Clifford algebra theory and/or Clifford analysis, or is it just a different name for the same set of mathematics? Or maybe does it provide a slightly different viewpoint on the same mathematics (like, e.g., probability theory having a completely different viewpoint from measure theory despite being formally measure theory)? If so, what are the advantages of this viewpoint?
2. Are there mathematical applications of "geometric algebra" outside the field itself? There seem to be applications to physics: are these applications mathematically rigorous?