There is no real difference between somebody and someone. Similarly, there is no difference between anybody and anyone, everybody and everyone or nobody and no one.

Note that the forms with body are a little more informal.

There is somebody at the door. (= There is someone at the door.)
Is there anybody? (= Is there anyone?)

Some- and any-

The difference between somebody and anybody, somewhere and anywhere and something and anything are the same as the difference between some and any.

The words somebody, something, somewhere etc., are used in affirmative clauses, whereas the words anybody, anything, anywhere etc., are used in negative and interrogative sentences.

There is somebody at the door.
Did anyone come?
I don’t want to go anywhere too expensive.
If you need anything just tell me.

When these words are used as subjects they are followed by singular verbs.

Everybody loves her.
Everything is ready.

Note that somebody is normally used to refer to only one person. To refer to more than one person, we use some people.

Somebody wants to meet you.
some people want to meet you.

To refer back to somebody, anybody etc., they, them and their are used with singular meanings.

Nobody came, did they? (Here the pronoun they refers back to nobody.)
Somebody left their umbrella in the office. (Here the pronoun their refers back to somebody.)