The difference is in the way they are tuned. The Special 20 is tuned closer to what's called "Just Intonation" (JI). In the old days, all harmonicas were tuned to JI. In JI tuned harps, chords sounded really smooth and nice. They were great for playing rhythms and chords.
However, 5 and 9 draw sounded flat in these harps. This wasn't a big issue when harmonica players played solo, but complaints arose when harmonica players started playing with other instruments. So, Hohner changed it's tuning until it got close to "Equal Temperament". Single notes sound good on equal temperament, but chords sound a little rough.
So, harmonica players started complaining again. Rick Epping (an amazing harmonica player and technician) came up with a solution by bringing the tuning closer to the original JI tuning, while maintaining a compromise to allow the Hohner Special 20 and Marine Band harmonicas to fit in with other instruments. Now the 5 and 9 draw are somewhat less flat. :)
The Golden Melody is tuned to Equal Temperament. So, it's great for melody playing, but not that good for playing chords. It's still a very good harmonica, and it will work with JP's lessons.