Persons would be more used in formal constructs, whereas people in more informal ones.

Grammar aside, peoples is the plural to mean "pessoas - nações - povos". Because you could use people meaning plural as well.  

That is, people meaning plural (pessoas, indivíduos), to know that it is plural you would locate some word that act as a signal that it is a plural. Take, for example the word some:

(algumas pessoas usam carros, mas nós vamos a pé.)

https://www.inglesnapontadalingua.com.br/2007/07/person-persons-people-peoples.html