Why do people use "peoples" and "persons" in some sentences? When is it right to use them in context?

No. Although both people and peoples are grammatically correct, they cannot be used interchangeably because they have different meanings.

Case 1: When you use it in the sense of one people, many peoples:

Here, people (singular) means a group of human beings belonging to a particular race/ethnicity. The plural of such groups is called peoples.

E. g.

The different peoples of the world have very diverse traditions. The Ukrainians are a nice people.

Case 2: When you use it in the sense of one person, many people:

Here, people means a group of persons. This is the regular and most common usage of the word people.

E. g.

There are more than a billion people in India