Well, as a working mathematician I find that the hard part is typically coming up with the right definitions. Once you have the right concepts clearly defined (in terms of things you already know), the actual new ideas tend to arise with ease.
Similarly, it is often the case that new fields of mathematics arise from someone defining a new concept that was previously imprecise. Once you have the right language to discuss something, discovering its properties becomes much more straightforward.
Mathematicians use and think about math in a way that is almost completely orthogonal to the way physicists use and think about it. To physicists math is more like a natural language, and definitions are conveniences. This is why mathematicians often say physicists can't do math: by their definition of math this is correct.
For physicists the math becomes a language and a safety net and a set of heuristics that let us simply the problem to the point of being about to reason about it effectively. A great deal of what we use math for amounts to book-keeping. The human imagination is as capable of dreaming up impossibilities as it is incapable of dreaming up the way the universe actually is, and math helps us avoid doing the former while we use systematic observation, controlled experiment and Bayesian inference to figure out the latter.
Because "thinking about the mathematical representation of physical reality" is such a profoundly unnatural, unintuitive act, and because the math is so strict and simple, it is very hard to for us to use it to imagine impossibilities, whereas if you have a conversation with a layperson you will find they almost instantly run off the rails into nonsense because they don't have the math to keep them on track. So laypeople believe in perpetual motion machines and the like with surprising ease, because they "just make sense" to their intuition (which maps pretty well to Aristotle's physics).
Similarly, it is often the case that new fields of mathematics arise from someone defining a new concept that was previously imprecise. Once you have the right language to discuss something, discovering its properties becomes much more straightforward.