The way I understand it, mathematical physics is not a domain of mathematics per se (contrary to, say, probability theory or algebraic geometry). It refers to areas of mathematics which are very strongly tied to theoretical physics. For example, studying the Navier-Stokes equation, or Schrödinger's equation with certain potential, is mathematical physics. Other areas of mathematical physics may be related to relativity theory, quantum field theory, etc.
The difference with physics is that, as usual in math, you may forget the physical motivation to explore different (more interesting, more general or on the contrary easier) settings, even if you lose most of the physical sense.
If you're interested, my advice would be : try to get a general overview of various fields in maths and physics (I don't know what your current degree of education is). Then, you'll have to decide if what you're really interested in is the physics or the maths (ask yourself, do I primarily want to understand how PDEs work, or do I want to understand how nature works ?) Depending on the answer, choose between math and physics. The choice of the field (string theory, field theory, relativity theory, etc.) will also depend on the mathematical tools used in this field, so be sure to build a large scientific culture instead of trying to choose to early.
As for mathematical logic, it's a completely unrelated question. Logic is concerned with questions such that : what happens if we replace such axiom by such axiom ? what is a proof exactly ? given a statement, is it always possible to find a proof of it or its negation if its false ?