Butterfly extension is a folding panel that's installed right into the table that unfolds when you extend the table, either manually or automatically. These tables don't require a separate insert.
Drop leaf extension involes sections of the table that fold up or down at either end. This is a very simple form of extension that doesn't require additional pieces.
Insert extension requires the use of an additional panel that the use inserts into the table. These tables can allow for more versatile extension, as multiple panels can more easily be accomodated than in a butterfly table.