The difference isn't about theorems, I think. Rather, its motivation: do you wish to do math for its own sake because you find it intriguing and elegant, or do you wish to do math because it can help people, solve real problems, and better our understanding of our universe (as opposed to idealized, abstract mathematical universes).
Personally I'm completely 100% in the former category, I could care less about the ramifications, but think the ideas themselves are beautiful.
I think most people in mathematics studying string theory, by the way, would put themselves as pure.
The University of Washington would seem to agree with me, here is what they say about the fact that they have separate departments for mathematics and applied mathematics, from their FAQ:
"The most important difference, perhaps, is that faculty members in the two departments are interested in a somewhat different range of subjects. A second important difference is one of motivation: Research in the Applied Math Department is a bit more oriented toward using mathematical ideas to solve problems that arise outside of mathematics, while research in the Math Department is directed more at understanding the underlying mathematical ideas, whether motivated by applications or by more theoretical considerations."