Correct usage of "persons" (vs. "people")

Generally, persons is a decent substitute for individuals, and appears more in legal contexts that demand precision.

People is the ordinary plural of person.

Asking for a table for two or a table for two people is better than asking for a table for two persons.

The NOAD on my Mac Mini had the following notes about using persons instead of people.

The words people and persons can both be used as the plural of person, but they are not used in exactly the same way. People is by far the more common of the two words and is used in most ordinary contexts: "a group of people"; "there were only about ten people"; "several thousand people have been rehoused". Persons, on the other hand, tends now to be restricted to official or formal contexts, as in "this vehicle is authorized to carry twenty persons"; "no persons admitted without a pass." In some contexts, persons, by pointing to the individual, may sound less friendly than people: "The number should not be disclosed to any unauthorized persons."

Similar notes can be found on Lexico (a collaboration with Oxford Dictionary hosted by Dictionary.com).

The normal plural of person is people, unless you're a bureaucrat.

According to this online article from DailyWritingTips, the words person and people are derived from different latin origins: persona and populum, respectively.

The article goes on to suggest that both person and people can be used in their own natural plural forms; persons and peoples, but modern daily use favours the following combination:

Singular: Person

Plural: People

The entry also backs up @onomatomaniak 's assertion that persons is used more for legal text, but doesn't indicate that persons is any more precise than people. The article doesn't reference any sources itself.