Mathematics is a social activity, in the sense that mathematicians create proofs to convince others that their findings are valid. Some proofs are easier to verify than others. Some proofs are "believed to be true" because no one has yet managed to prove them wrong yet.

So if you think about "real world usage", you can either use the math results and implicitly trust that they "just work", or you can dive a bit deeper, see if you agree with those results, or at least gain some insights from the proofs that have been presented.

And just to be clear, there is little to no "real world usage" math without academic math.