Question: What would it mean to say that mathematics was invented and how would this be different from saying mathematics was discovered? 

Is this even a serious philosophical question or just a meaningless/tautological linguistic ambiguity?

Answer: 
This is a serious question and it is the same as saying: is the knowledge in mathematics universal or a human construct?

Pi (the number, regardless of its base) and many other things are universals, mathematics are discovered to that extent. Then they can be used to formalise inventions that may prove to be wrong, right or paradoxical, in the same way the knowledge (discovered) about horses and rhinos can be used to (invent and) speak about unicorns (that were never discovered).

Can we say (as many answers point here) that biology was invented because of unicorns?
Mathematics is normative. That is clear when one reads Euclid and Lobachevsky in juxtaposition, or Euclid and Descartes, or Euclid and Leibniz or Newton, or Leibniz and Newton and Dedekind, or Dedekind and Canton, or Canton and Godel, etc., etc.. Geometry is clearly normative, as we have different geometries (although one might claim, "yes, but they can all be transformed into one another"). But the argument goes like this: there is no other arithmetic; and thus, in counting (and its extensions), we are discovering something fundamental to the universe. Of course, such an answer supposes that Euclid and Dedekind are talking about the *same* arithmetic. Is that even possible? No. There's no room, in Euclid's conception of number (think of Books V and VI of the *Elements*), for Dedekind's cuts, and thus, no room for a whole host of numbers that are incompatible with Euclid's concept of number. And if you think that the concept of number is fundamental to a conception of arithmetic, then it would seem that every time we "add" new "sorts" of numbers (which are invented by new sorts of functions), we create a new arithmetic. But, someone might say, "that's all well and good, but we really just subsume those other arithmetics under what we call arithmetic--there's really just one arithmetic." But that would be like saying "wave-mechanics really just subsumed ordinary mechanics...." Such a statement doesn't make any sense.