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: 
My personal point of view is that mathematicians *invented* the axioms and the rules of operation, the rest are *discovered*. Mathematicians *invented* the notations for writing down the concepts which are *discovered* within the universe of an axiom.

The concept of numbers exists, but we invent the notation that the glyph '1' and the sound /wʌn/ refers to the concept of singular object that we discovered. We invented the rules of matrix multiplication, but the consequences of the way we do matrix multiplications are discovered.

Most of the time, we deliberately invent a set of axioms that will lead us to discover a set of facts we want to be true. This is certainly true with imaginary numbers, we invented them so that we can discover the solutions to problems we previously were unable or difficult to solve.
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.