Going back to the Latin, "person" derives from "persona", which refers to an individual. "People", on the other hand, comes from "populum", and it refers to a group of persons sharing a culture or social environment. So in origin, "person" is a singular word with the plural "persons", and "people" is a singular word meaning "a group of miscellaneous persons" and having the plural "peoples".
Neither OED nor Merriam-Webster gives "people" as the plural of "person", even informally. However, many writers have used people as the plural form of person, and nowadays it is widely accepted. Notice that legal and very formal texts still use persons as the plural form.
It would certainly sound wrong to describe the trinitarian Christian God as "three people" rather than "three persons".