Mathematical physics is physics! Mathematical physics is a branch of physics which has solid mathematical foundations and researcher in this area have a high expertise in mathematics. They understand advanced mathematical concepts which they use regularly in their research, but they don't pay much attention to rigor or proofs. The motivation for a mathematical physicist is to study physical systems using advanced mathematics and which sometimes result in predictions about the mathematical objects (underlying the physical problem) under study. These predictions become conjectures in mathematics which are left for mathematicians to study rigorously and prove. Many a times, mathematicians studying the results of physicists come up with more theories and generalizations. Mathematical physicists study the physics implied by these new results. So it's kind of a symbiotic relation. A good example of mathematical physics is the theory of general relativity which was built on the mathematics of Riemann. In modern days string theory, supersymmetry, topological quantum field theories, Ising models, loop quantum gravity etc. are famous areas of mathematical physics. Let me give you another example from the work of Edward Witten (a mathematical physicist who was awarded the Fields medal). Jones polynomial is a polynomial invariant of an oriented knot (Knot theory is an area of topology). Vaughn Jones received a Fields medal for this work. Edward Witten showed that the Jones polynomial can be obtained by considering ChernSimons gauge theory on the three-sphere. Now Witten didn't prove any mathematical results, but he showed that the same polynomial also arises in a physical system. Later Khovanov defined another invariant of oriented knots called Khovanov homology which is a generalization of Jones polynomial. This opened a questions for physicists to understand the physical meaning of Khovanov homology in the context of gauge theories. Dunfield-Gukov-Rassmusen and Witten independently approached this problem and explained Khovanov theory in physics.