One application that I know of the Mumford conjecture is Teleman's proof of Givental's conjecture in this paper. Givental's conjecture states that when the quantum cohomology of a smooth projective variety (or compact symplectic manifold) is (generically) semisimple (meaning that the algebra is semisimple for generic values of the deformation parameter), then the higher genus Gromov-Witten invariants are uniquely and explicitly determined by the quantum cohomology. Since quantum cohomology consists of genus 0 information, another way to say this is that the higher genus Gromov-Witten invariants are uniquely and explicitly determined by the genus 0 Gromov-Witten invariants, when the quantum cohomology is semisimple.
The proof, extremely roughly, uses the Mumford conjecture in the following way: semisimplicity allows you to go back and forth between low genus and high genus without loss of information. The Mumford conjecture tells you what things look like in very high genus, so if you want to know what something in some arbitrary genus looks like, you kick it up to very high genus, identify it by the Mumford conjecture, and then kick it back down to its original genus.
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You might then ask, which smooth projective varieties (or compact symplectic manifolds) have semisimple quantum cohomology? Smooth projective toric varieties form a class of examples. Arend Bayer showed that semisimple quantum cohomology is preserved under blowing up points. There are other examples...
There is an interesting conjecture of Dubrovin which states that a variety has semisimple quantum cohomology if and only if its derived category has a full exceptional collection. (This conjecture is based heavily on mirror symmetry philosophy...) I don't know the status of this conjecture. But I think it is true, for example, that having a full exceptional collection is preserved under blowing up points. (Proved by Orlov? Bondal?)
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It is also apparently possible to view the Mumford conjecture as a special case of the cobordism hypothesis. Take a look at Jacob Lurie's paper on TFTs, particularly section 2.5.