In general, the difference is so subtle that it doesn't matter.
I agree with this answer more than with mine. I leave my answer here for the sake of public record (and because I can't delete an accepted answer).
My original answer below:
"Anyone" implies that the universe of people is not everyone on Earth. Usually, based on context, "everyone" also means that, but it's not inherent in the word.
"Tonight after the meeting, anyone is welcome to ask a question of our panelists"
implies that it's only the people at the meeting.
"Tonight after the meeting, everyone is welcome to ask a question of our panelists"
is only a very bit awkward since it implies that people can come in from another place entirely and ask a question.
On the other hand,
Everyone is welcome to come!
Anyone is welcome to come!
on the other hand, implies:
Anyone who wants to is welcome to come.