Well there is a fine border between the two, which includes, as an example, quantum field theory. QFT is a very strong physical theory, but needs a lot of backing up in terms of rigorous mathematics. Topological quantum field theory is something I just recently read about and is more of the mathematics side of QFT, whereas as QFT is more physical. But they share ideas, concepts, and tools. They play off of one another. Physicists readily use the Feynman path integral, but to my knowledge, has not been shown to be a completely rigorous technique, especially compared to the mathematical techniques applied in quantum mechanics, which are very rigorous. So there is an example of the physicists using something that was introduced physically, but can be bolstered mathematically, giving even more credit to the physical theories it's applied to. They are different fields, but are very closely related, at least on the borders of theoretical physics.
Physics is really very special in that it can attack problems in two very distinct ways. It has experimental techniques, and it has theoretical techniques. Where one falters, the other can pick up until the other catches back up. It's really a fantastic interplay. For general relativity, the mathematics came first, which was then bolstered with experimentation. For the Higgs boson, it has been predicted mathematically, but is waiting for the experimentation. Things like Heisenberg's uncertainly principle was sort of discovered on a physical basis, but now there are strict and rigorous mathematical proofs of the uncertainty principles. Faraday and others observed the interplay between electricity and magnetism experimentally and then the observations were supplemented by the mathematics developed later on which came to the discovery of Maxwell's equations. Maxwell and his peers originally had somewhere around 23 equations, which were whittled down to 4. Now, using the mathematical development of differential forms, Maxwell's 4 equations can now be stated in extremely simple terms in just 2 equations. This discussion could go on and on, as the interplay between math and physics is fundamental. Even David Hilbert, a great mathematician and theoretical physicist who independently discovered general relativity, suggested as one of his 23 problems to provide an axiomatic development of physics, similar to what was done for mathematics in the 18th and 19th centuries.
. I think physicists in most cases would have a much harder time doing mathematics than mathematicians doing physics, without getting giggled at...
