Both of these words refer to people who work at a company - but staff is always singular and uncountable - it describes the entire group of workers as one thing. Employees is plural and countable - it describes the collection of individual workers.
The entire staff was happy about the extra day off.
All the employees were happy about the extra day off.
Note that "staff" takes the singular verb was, and "employees" takes the plural verb were.
Whenever you use a specific number, use employees or staff members:
Basically the real version of the doomsday bunker where you always see the President go in the movies, it is 48 miles outside of Washington DC in the Blue Ridge Mountains. Not much has been said of the place apart from a short TIME magazine interview of one of its retired engineers.