Typically, the word "crew" is used when speaking about a group of people who perform jobs that require manual labor. For example, in an office, the people who clean the office may be called a "cleaning crew", but the workers in clerical, operational, and management roles would not be. Staff and personnel can be used when talking about any group of employees.

It's all pretty interchangeable. "Team," as well. In car racing, you have a "pit crew". Different restaurants and stores might have different preferred vernaculars, but "staff" is the most obvious term for a group of workers and could be used comfortably in most situations.