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:
The best thing you can do for your skin at the start of summer is not to over-dry it and not over moisturize it either, says Dr. Gross. If you use a heavy moisturizer, switch to a light, daily source SPF. If your skins oiler, find an oil-free sunscreen stick.