Having a deep (proof-based) understanding leads to more "real world" insights. So, there may not be a clear distinction.

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.

> create proofs to convince others

Well, it is not only about people. You would also want to "convince" Nature (so that the aircraft you designed does not crash right after a takeoff), your own or, often, someone else's wallet (so that money is not spent on nothing).

But the distinction is generally not as clear as you may think. (1) Much of the mathematics came from real world problems (so, in particular, one may get drawn into some kind of mathematical research and even end up discovering new mathematical facts). (2) Sometimes when applying mathematics one still needs to employ deduction (derive a formula, prove a statement one wants to rely on, etc.).