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I'm a huge fan of mathematical physics and I know what the formal definitions of those two areas are, I've seen them. But I still get completely baffled when someone asks me to explain it simply. The difference is obvious to me, but I just can't seem to put it into words in a satisfying enough manner. So I'm asking you to help me...

What is the essential difference between theoretical physics and mathematical physics?

or if you prefer the rephrased version...

What was the motivation for introducing the name "mathematical physics" as a separate entity?

(It doesn't matter if you're one of those people who don't like labels, the thing is that we do have these two separated very often in academia.)

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Theoretical physics is the field that develops theories about how nature operates. It is fundamentally physics, in that the ultimate goal is to describe reality. It is informed by experiment, and at the same time it extends the results of experiments, making predictions about what has not been physically tested. This is accomplished using the language of mathematics, and often the demands of theoretical physicists force mathematicians to extend this language in new directions, but it is not concerned with developing the language of math. Theoretical physicists are, among other things, physicists who are very well-versed in math (which is not to say other physicists are not - please don't hurt me).

Mathematical physics, on the other hand, is a branch of mathematics. It explores relations between abstract concepts, proves certain results contingent upon certain hypotheses, and establishes an interlinked set of tools that can be used to study anything that happens to match the relations and hypotheses on hand. This branch in particular is motivated by the theories used in physics. It may seek to prove certain truths that were simply assumed by physicists, or carefully delineate the conditions under which certain theories hold, or even provide generally applicable tools to physicists, who can in turn apply them to nature. Mathematical physicists are mathematicians who are intrigued/inspired by physics.

One could say that mathematical physics is concerned with the internal, logical consistency of physical theories, while theoretical physics is concerned with finding the right model to describe the world around us. Very roughly, one might diagram these things as shown below. $$ \text{Mathematical physics} \Longleftrightarrow \text{Theoretical physics} \Longleftrightarrow \text{Experimental physics} $$

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    $\begingroup$ I like this answer, but I don't like to think of mathematical physics as strictly a branch of mathematics. I think there are mathematical physicists who are physicists; ie I think the first and last sentences of your definition of 'mathematical physics' are misleading. $\endgroup$
    – levitopher
    Commented Mar 9, 2013 at 4:28
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[Converted from my comments, as suggested]

Mathematical physics and theoretical physics are two very distinct disciplines, as can be checked by browsing a random issue of Communications in Mathematical Physics. Mathematical Physics is bona fide mathematics, but applied to physics questions: the papers have the traditional Lemma/Proposition/Theorem structure of mathematics papers, which is very different from that of a typical paper in theoretical physics. And it's not just a matter of style: papers in mathematical physics often could just as well be published in mathematics journals, which is not the case for almost all papers in theoretical physics.

The approaches are also very different. Theoretical physicists are much more pragmatic, and consequently allow themselves all kind of uncontrolled approximations, mathematically meaningless operations, etc. In mathematical physics, once the problem is posed, the derivation is supposed to be purely deductive, without any additional, implicit assumption.

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    $\begingroup$ Let me point out that this does not correspond to an actual difference in the subject matter studied by theoretical physicists and mathematical physicists; just the ways in which they study it. $\endgroup$
    – Peter Shor
    Commented Mar 9, 2013 at 13:27
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    $\begingroup$ @PeterShor: Yes, that's right, although, there is usually a temporal shift: topics studied by mathematical physicists have often been considered well understood for years by theoretical physicists ;) . $\endgroup$
    – Yvan Velenik
    Commented Mar 9, 2013 at 14:01
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    $\begingroup$ @DIMension10: Sure, there are various degrees of rigor, depending on the subfield. String theory has a quite distinct status, as it is often not fully rigorous, but yields many interesting conjectures. This entire field lies somewhere between Math (because of the advanced math. framework required, and the many fascinating predictions, etc.) and Physics (due to lack of direct experimental verification). But the situation is very different for the other fields. In most (all?) other fields, CMP publishes fully rigorous works. This is the case, for example, in my field, statistical physics. $\endgroup$
    – Yvan Velenik
    Commented Nov 2, 2013 at 8:26
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    $\begingroup$ @DIMension10: The particular status of String Theory can also be seen in the fact that Witten has received the Fields medal, that string theorists regularly give conferences at the International Congress of Mathematicians, and that many math. departments have people working in this field. This shows that one can contribute (in important ways) to mathematics without necessarily doing rigorous work. Of course, not all mathematicians are happy with this kind of semi-rigorous maths. I don't have any problem with that myself, but I come from physics, after all... $\endgroup$
    – Yvan Velenik
    Commented Nov 2, 2013 at 8:32
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    $\begingroup$ @Yvan: as far as I can tell, Witten did not receive the Fields medal for string theory. The citation said merely "proof in 1981 of the positive energy theorem in general relativity". The honorarium lecture gives details on this, rigidity theorems inspired by string theory, and topological quantum field theory. For the first, Witten gave a complete proof. For the second and third, Witten's work gave the tools that let other mathematicians give a complete proof. $\endgroup$
    – Peter Shor
    Commented Feb 15, 2016 at 12:53
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Mathematical physics is what is done by mathematical physicists, and published in mathematical physics journals. Theoretical physics is what is done by theoretical physicists, and published in physics journals.

I believe the actual dividing line between the two fields was delineated largely by the historical development of these fields. Certainly, theoretical physicists are somewhat more interested in connecting with physics experiments, and mathematical physicists are somewhat more interested in connected with other fields of mathematics. But any actual conceptual boundary between the two disciplines is very fuzzy.

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I'd say simply that mathematical physicists are interested in mathematical applications, while theoretical physicists are interested in physical applications. Many physicists are some combination of the two.

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Theoretical Physicists figure out how nature is described by scientific language, or mathematical language. And mathematical physicists develop mathematical tools in order to provide carrier of physics science.

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Focus (categorical closure, or attempt at a closure): Current physical theories, but formulated strictly, and as rigourously/unambiguously as possible, mathematically. Plausible mathematical models of physical systems for which a satisfactory theory does not yet exist. Extrapolations of such theories or models that could be of interest to physics.

Come to think of it, this may include almost everything mathematical, so almost insatiable curiosity about mathematics, at the very least, is a must.

Techniques & Concepts (toolkit): Those of mathematicians: continuity/differentiability, mappings, dimension, convergence, symmetry, and the like.

Focus on particular techniques and concepts: Symmetry and invariance are paramount. You just can't do without.

Background: You'd better have a solid background either as a mathematician or as a theoretical physicist, and grope your way relentlessly towards the brethren discipline.

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