I'll stick to sequels where I thought the original was significantly worse.
Lost Kingdoms 2 (Rune 2) - Got rid of instanced battles, got rid of the stupid discard mechanic, added badly-needed card variety, added lots of cards and the Mech type, added transform cards. Unfortunately, Banshees became the absolute most broken attack/item/thing in any video game ever.
Sonic 3 & Knuckles - Added a real narrative to the game, introduced a 3rd playable character with his own routes and story, doubled the number of bosses (per level, that is), added more shields, greatly lengthed the levels, introduced deeper gameplay with more interact-able objects, and most importantly, ADDED SAVING!
Paper Mario: The Thousand-Year Door - (I'm gonna get so much shit for saying that.) The N64-to-Gamecube difference totally improved gameplay, making the action commands actually work properly. Also, the badge and star power systems were greatly improved, but made the game a little too broken.
Sonic Adventure 2 (and SA2B) - I don't need to spell it out.