What is the difference between an em-dash/double-hyphen and an ellipsis?

Both em-dashes (–) and ellipses (…) are used to indicate unfinished thoughts, but their uses differ slightly.

Em-dashes are used to indicate a thought that ends abruptly, either because the speaker ended it that way for effect (“aposiopesis”), or because he or she was interrupted.

THOMAS

I was just about to tell your dad that our math test was --

FRANK

-- Printed in green ink! Can you believe that? Green ink? For a math test?!

Ellipses are used to indicate a thought that trails off.

FRANK’S DAD

And you’re telling me this because...?

Both marks have other uses as well. For example, em-dashes are also used to set off certain dependent clauses, and ellipses are also used to indicate that a part of a quote has been omitted.