Okay, then what's the difference between Feuer and Brand, *and* what's the difference between Wand and Mauer?
Feuer is the more primitive, more general term. Brand is derived from the verb brennen, "to burn", and is mostly used for a destructive, accidental fire. I.e. a bushfire is a Waldbrand (forest burn), you can get insurance against Hausbrand (house burn); but the campfire you roast marshmallows on would be a Lagerfeuer; the fire in a stove is a Herdfeuer, one in a fireplace a Kaminfeuer. (Kamin is a chimney.)
A Wand refers to the geometric configuration that delimits space, Mauer to the building of walls from stones (mauern is the verb; a mason is a Maurer. It's probably related to mural.) Not every Wand is made out of stone: it could be made from paper, or just a cardboard dividers used to separate cubicles; cells in an organic body are separated by Zellwände (cell walls); a bulletin board is a Pinnwand.
A Mauer is very likely made from stone; mauern is also used figuratively to mean "blocking, not talking";
German speakers might be familiar with a children's tune that starts
The strongest implication of the use of Brandmauer rather than Feuerwand is that the word is one that originated in architect's jargon; I find that, at least in German jargon, construction-based and more specific references are more likely than general ones.
The word Feuerwand, although uncommon, could be used in an article about the work of firemen (Feuerwehrmänner - not Brandwehrmänner, I have no idea why) to describe a wall consisting of fire.