Some important applications of Teichmuller theory are to 3-D topology
(I mean the work of Thurston, and later development of this work), and to
holomorphic dynamics. On this I refer on the books by Thurston and by Hubbard,
and on the paper of Douady and Hubbard A proof of Thurston's topological characterization of rational functions, MR1251582, but this was already mentioned in the original question.
Actually, since the work of Sullivan, Douady and Hubbard, Teichmuller theory is
one of the main tools in holomorphic dynamics.

There are also applications to string theory, and I expect that some string theorist will write about them. But here is a nice example, where string theorists made a pure mathematical conjecture, and it was proved using Teichmuller theory:
MR0882831 Zograf, P. G.; Takhtadzhyan, L. A. On the Liouville equation, accessory parameters and the geometry of Teichmüller space for Riemann surfaces of genus 0.