Both staffing and recruiting are used to hire candidates for a particular role. However, staffing is commonly used to fill short term roles or to hire people for specific projects. And, because of this, filling these roles tends to be a rather swift process.
On the other hand, recruiting tends to take on the task of finding a more permanent role. That means it might take longer to fill searches simply because it’s necessary to be more strategic and detailed to find a person who has the proper qualifications but is also a fit culturally.
In terms of working with individuals/potential candidates, staffing agencies/firms will generally work with those who are seeking jobs. On the other hand, recruiting firms deal with those who are seeking jobs as well as those that are not. This is commonly referred to as active candidates and passive candidates.