Cantor was very troubled by his discovery of the transfinite numbers because they conflicted with the infinite nature of God (hence the name transfinite).
Shouldn't be any surprise to anyone that when a theory conflicts with _beliefs_ there's trouble.
They use the term 'discover,' I would presume, because it's the most technically accurate term for what they do. As near as I can tell, as a relative layperson, mathematicians look at the existing state of mathematics, embodied in proven theorems, and consider the implications of those theories, looking for unanswered questions or outright contradictions. They then attempt to find the new theory that will provably answer the question or resolve the contradiction.
The only 'invention' that goes on seems, to me, to be on the order of brainstorming, i.e., inventing hypotheticals to test their implications or shed new light on an existing conundrum.
Fundamentally, I think the use of 'discovery' terminology, rather than 'invention' terminology arises from the nature of the new thing produced. Discoveries, in the scientific or mathematical sense, were 'always there' within the corresponding realm of inquiry (e.g., the set of axioms that comprise mathematics, the physical world for scientific discoveries, & etc.) and represent a mere formulation of an existing but previously hidden truth. Inventions, on the other hand, represent a novel application of existing principles for some external purpose. The implementation cannot be said to have flowed from existing knowledge in any meaningful way. Nor do inventions have any general theoretical utility. (Mousetraps are fun little inventions, but don't contribute to a 'Theory of Pest Control' and we certainly would not refer to the 'discoverer of the mouse trap.')
Additionally, other scientists do in fact use the terms 'discover' and 'invent' in precisely the same sense as mathematicians do, and for similar reasons. In fact, these are the commonly accepted notions of the terms even among nonspecialists.
Using 'mathematical invention' rather than 'mathematical discovery' erroneously (or maliciously) ascribes arbitrary subjectivity to mathematical thought.
they say "discover" because they believe it was present ever since. just like physics don't say they invented helium or oxygen, they just discovered it.
Shouldn't be any surprise to anyone that when a theory conflicts with _beliefs_ there's trouble.