The Special Edition means that all graphics of the game were redone, so everything from Guybrush to the backdrops of the island have a new, modern visual flair to them. The modernized look has more than a passing resemblance to the style used in the Telltale Tales of Monkey Island series, which is nice for gamers who aren't as familiar with the series' older titles. The other feature that makes the graphics of the Special Edition stand out is the ability to toggle between the new visuals and the classic pixelated graphics.
Flipping between the modern and classic styles does more than change the graphics though. The game reverts back to its roots so much that the new audio (both reworked music and new dialogue) goes away. That's right, not only was the game's original soundtrack redone to be a much richer sound, but the voice actors used in later titles were called in to read the dialogue from the original game, and both updates to the game's audio sound great.