It's like deja vu all over again.
Basically Castlevania III is the first game on a larger scale. More challenging levels, multi-part bosses, additional characters that you can choose to play as, and on the whole an improved game all around. But why are the controls still horrible?! Well, they're not terrible but they've remained fairly unimproved. Traversing stairs is still a nightmare, it's easy to fall to your death. Also the jumping still feels slightly off, as you somewhat 'slide'; it doesn't feel precise.
Where they've really altered this game is the level of challenge. Some of the sequences are quite difficult, such as trying to maneuver jumps whilst dealing with a horde of enemies. Although this was present in Castlevania I, not in such a horrific and endless torrent. The bosses are multi-stage which can also add to the difficulty if you're low on health. As for Death and Dracula, ouch. Words can't express how frustrating these bosses are.
The story is slightly more fleshed out in Castlevania III but it still doesn't really amount to much, not surprising for an NES game. If you were playing in hopes of getting a great ending then you'll be disappointed. However, this a great addition to a classic series, one of few the games I've played on the system that stands the test of time well.