When is the proper time to use the word "persons" in a sentence instead of using "people"?
Michael Swan (Practical English Usage, 2005.524) notes "Persons is sometimes used as a plural of person in official language" (my emphasis added). This supports my own view that we need never use persons in our own personal or business life. Leave it for the lawyers and law-makers to write such titles as The Value Added Tax (Vehicles Designed or Adapted for Handicapped Persons) Order 2001 (found here).
I can't provide any sources (at least, not yet) but you might find this anecdote interesting. I can't vouch for its accuracy, but it does make sense.
My wife is a hotelier, currently as a General Manager, and in that industry they often use persons. Naturally, I asked why. The answer I got is that persons is used for individuals. "Twenty persons" is twenty individuals who don't know each other. A single group of twenty would be described as twenty people, but the 'people' is usually dropped so it is "a group of twenty". The reason is that groups and individuals are processed differently so they need different terms.