Word 2013 has added a whole new layer to the difficulty of creating a new document based on a custom template (or at least added extra steps).

File New Variations in the Versions of Word

You need to use the File > New command and go to your Shared or Personal templates, or use the classic FileNew dialog.

You do not apply a template, you use it to create a new document. You can do this through File > New or by double-clicking on the template.
Templates in Microsoft Word