People at my company see unit testing as a lot of extra work, that offers fewer benefits than existing functional tests. Are unit and integration tests worth it? Note a large existing codebase that wasn't designed with testing in mind.
(I'm assuming that you're using "functional test" to mean a test involving the whole system or application being up and running.)
I would unit test new functionality as I wrote it, for three reasons:
(This includes bug fixes, as suggested by Epaga.)
I would strongly recommend Michael Feathers' "Working Effectively with Legacy Code" to give you tips on how to start unit testing a codebase which wasn't designed for it.