Choosing between "everybody" and "everyone"

When addressing a radio audience, is welcome everybody the appropriate word choice? Why not welcome everyone?

What you have in mind should be written as:

Welcome, everybody!

Which is equivalent to, for example:

Welcome, Janet!

Without the comma as a sentence, it would be, for example:

Janet, go and welcome everybody so they understand the party has already started.

As whether everybody and everyone are interchangeable - yes, they are. It doesn't matter which one you use in this case.

Both forms are valid, but Welcome everyone is more common as a "collective greeting".

When reporting what the announcer did, you might say he Welcomed everybody, meaning he welcomed the entire audience collectively. But the announcer himself is more likely to say "Welcome, everyone!" because this conveys more the sense of simultaneously addressing each person individually, making each listener feel more intimately/personally involved.

I can't say for sure, but I think a stand-up comic might be more likely to start his act with "Welcome, everybody!" because he'd be more concerned with encouraging the audience to think of themselves as a collective body (who'll encourage each other to laugh at the right moments, for example). That doesn't apply to a radio show, where the audience have no feedback between themselves. The radio show host is more concerned with drawing each listener in as an individual.