Benefits to you
You're already paying a full-time staff. We can assume they're good at their jobs. Why should you take on the additional expense of a consultant?
Fair enough. My answer:
Your staff is already busy, fully loaded with projects and daily tasks that you need done.
When someone takes on a new project, someone else will have to do what they were doing. The substitute will probably not be as effective at first.
When the problem is solved or the project is finished, you have to reassign people again to keep them productive.
You may need a fresh perspective. You can't get much fresher than someone outside your company.
You may find just the right person but she doesn't want to relocate.
It takes time and money to recruit. Sometimes you need additional resources right now.
You may need to acquire a different skill set for an unusual challenge, or maintain a skill set on retainer to service an irregular business segment, but can't justify a permanent position.
That's where the right consultant presents an opportunity.
When choosing a consultant, make sure he or she can communicate clearly at all levels; can get along well with management, the shop people, and customers; knows his or her limitations; is impeccably honest and discreet; and has the skill set you need.
Bottom line: I offer what I do best in the photonics industry, to those who can see its value, when they want it, and then I leave without attachments. I know whether I know.
And if I don't know, I usually know someone who does.