They know the rules, and can also offer advice
Like a good accountant, a good CFPS not only knows the bean-counting rules, they also know a lot of general rules, and a couple of "magic formulas", and they're willing to share that information with you.
I can tell you from personal experience, I've worked with several accountants over the years, and the guy that just does my books is okay, but the guy that tells me beforehand, "Al, you don't want to do that", that's a guy who is very valuable. I've written here before, I like one of my lawyers, but I love my tax lawyer. I actually save money by talking to that guy.
This isn't their first project
My personal opinion is that a good CFPS will have worked as an apprentice with a good CFPS, and will have many projects under his belt long before you begin working with him. A lot of software projects are similar, but there are also significant differences in them as well, and you really want a person who has seen a lot of different projects before they work on your project.
And again, the experienced person should be able to give you other advice besides counting function points, while someone new to FPA will just be trying to figure out how to make an accurate count.
This is someone you feel comfortable working with
Finally, as with any other service, you need to feel comfortable with the person offering their service to you. Only you can decide whether they seem trustworthy, and someone who will work well with your team.
For example, I talked to a lot of lawyers over the years before I found a couple that I really liked, and I had similar experiences with accountants, and auto mechanics. Whatever the reasons are, with some people you "click" right away, while other people give you "bad vibes", and again, that's your call to make.