This is a question that naturally arises as I consider myself to be a mathematical physicist, so I do mathematical physics. But what is mathematical physics?
I don’t think there is any fully agreed on definition of mathematical physics and like any branch of mathematics and physics it evolves and grows. That said, there is roughly two common themes:
That is trying to apply mathematical rigour in the constructions and calculations of physics. This is often very hard as physics often requires lots of simplifications and approximations. A lot of physical interpretation and intuition can enter into the work. Physics for the most part is not mathematics and lots of results in theoretical physics lack the rigour required by mathematicians.
Mathematics is the framework in which one constructs physical theories of nature. As such mathematics, as mathematics is fundamental in developing our understanding of the world around us. This part mathematical physics is about studying the basic structures behind physics, often with little or no direct reference to a specific physical systems. This can lead to natural generalisations of the mathematical structures encountered and give a wider framework to understand physics.
We see that mathematical physics is often closer to mathematics than physics. I see it as physically motivated mathematics , though this motivation is often very technical.
Of course this overlaps to some extent with theoretical physics. However, the motivation for theoretical physics is to create and explore physical models, hopefully linking them with reality. Mathematical physics is more concerned with the mathematical structures. Both I think are important and feed of each other a lot. Without development in mathematical physics, theoretical physics would have less mathematical structure and without theoretical physics, mathematical physics would lack inspiration.