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Everyone and everybody are synonymous. They are used in almost identical contexts with identical meaning. A native speaker has probably never given a thought as to the choice of these two words; we use them with equal frequency.
There is a usage question that is commonly asked about these words. They are both words that identify a group of people but take singular verbs:
Everybody on the team goes to bed at 10.
Everyone in the office likes coffee.
What pronoun is used after everybody and everyone? The words are grammatically singular but plural in meaning. The answer is they, their, and them:
Everybody on the team goes to bed at 10 because they get up early.
Everyone in the office likes coffee but some of them bring their own.
You can also use his and her if you want to emphasize the individuals in a group:
Everybody does his or her work alone.
Everyone has his or her own problems.
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The short answer is, there's not much difference! Both of these words mean "every person," and in dictionaries, the meaning of everyone is often given as everybody, and vice versa.
However, it's worth mentioning that many people think everybody is a little more casual (more informal) than everyone. Also, everybody is used more often than everyone in spoken language, which makes sense if it's more informal. Having said this, it's absolutely fine to use either one. You have a choice.
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Also:
Usage: Everyone and everybody are interchangeable, and can be used as synonyms of each other in any context. Care should be taken, however, to distinguish between everyone as a single word and every one as two words, the latter form correctly being used to refer to each individual person or thing in a particular group: every one of them is wrong.
http://www.thefreedictionary.com/everyone