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Everyone : we use this word to describe all the people you are talking about. That is, they have the same identity, goal or status.
Example : Everyone is happy that Mary is healthy again. (Here everyone has the same feeling - happy. No one is different.)
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Each one : Each one is to describe one of a group (more than 1 person) of people. They may not have same identity, goal or staus.
Example : Each one in our school played an important role in making the school open day a success. (Here each one can refer to the principal, the teachers, the students or the parents. Each one of them is the unit of the group (which organize the open day) but they are different in identity or status.)