The successful historical relationship between Mathematics and Physics seems to be when there is some pre-existing Mathematical Theory M (e.g. Group Theory, Topology, Non-Euclidean Geometry - or special cases of these) and an evolving Physical theory P. P might be expressed in a different kind of mathematics, if at all. Then a mapping is found which maps P onto M, except that M has more equations or components than this mapping includes. So the question becomes:
"Do the missing components of M map onto some (so far) unobserved properties of P?"
When this is successful (as in many of the other examples cited) then historians say that mathematics has proven valuable for Physics (yet again).
Another example of this phenomenon might be the SU(3) classification of some particles which had a gap in the representation when mapped onto known particles; the gap mapped onto the $\Omega$ particle. Didnt Gell-Mann get a Nobel prize for that one?
String Theory seems a little bit different from this classic scenario (but it is not the only such) where there is a deliberate attempt to develop Mathematics to model known (and perhaps unknown) Physics. This might be seen as a form of anticipation.
Some mathematicians take a stronger view than the account given here, in that they believe that the Physical Universe is fundamentally mathematical in character. Often, like Penrose, they may have specific types of mathematics in mind with that claim. So from this perspective the development of that mathematics is valuable over and above any current experimental data. A similar belief seems to underlay the String Theory efforts.