Some other 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.

There are also applications to string theory, but let some string theorist write about them.