What is the difference between a boat and a submarine?

Nothing - the Word "Boat" has been applied to Submarines since they were first used in the Revolutionary War, and the term has stuck with them since then.

A "boat" in Naval parlance is any vessel that is launched from another ship or tender, such as a launch, whaleboat, etc. Early submarines required a support vessel (tender) to launch and recover them;though modern submarines don't require such, they do need support from tenders for supplies and repairs.

Submariners always refer to their ships as boats, and not ships. The reason for this is that to submariners there are only 2 kinds of ships; Submarines, and Targets.