SSB -> Melee
DQ7 -> DQ8
Zelda 2 -> Link to the Past
SMB2 -> SMB3
Pokemon R/B -> G/S

Batman Arkham City.

Legend Of Zelda Skyward Sword > Twilight Princess
Smash Bros for Wii U > Brawl
Fatal Frame 2: Crimson Butterfly > Fatal Frame
Luigi's Mansion Dark Moon > Luigi's Mansion

Ratchet and Clank 2016

Hitman Blood Money
Heroes of Might & Magic II
SSX3
Flatout 2
Dirt 2
Eyyup 2

Talk about winning a thread on the first post haha, absolute garbage to all time classic.

Here's a couple:

Ultima VIII -> Ultima Online: I don't know if this is really fair as they're not exactly of the same series, but Ultima essentially went from their weakest installment that looked like it might end the series to one of the greatest and most enduring MMO's of all time that people are still playing 19 years later.

GTA II -> GTA III: I think this one should be self-explanatory, probably one of the biggest upgrades and most important reinventions of a series in gaming history.

Europa Universalis II -> Europa Universalis III: I know it's probably not as well known, but the leap that grand strategy franchise took was enormous.

Sonic the Hedgehog -> Sonic the Hedgehog 2: The first game was certainly good in its own right, but I don't think modern Sonic (with the spin and everything) is recognizable until the sequel.

Dynasty Warriors -> Dynasty Warriors 2: Even though it's akin to Madden with little changing from iteration to iteration, this franchise has always been a guilty pleasure of mine. The original was merely a dueling game between leaders, whereas the second game reinvented itself with the huge maps and thousands of soldiers.

Pitfall -> Pitfall 2: Don't get me wrong, the first Pitfall is among the best 2nd gen games in its own right, but to this day I have a hard time understanding how Pitfall 2 even runs on an Atari VCS. Amazing game given the tech limitations.

Call of Duty -> Call of Duty 2: Whereas Call of Duty was, to put it simply, "meh", CoD 2 is the game that kept all my friends and I busy in the dorms and really launched that franchise.

Metroid -> Super Metroid: Metroid is a classic, but they basically upgraded every aspect of the game for Super Metroid.

Uncharted 2 over 1

Uncharted 2 and Killzone 2 come to mind, especially the latter was a major improvement.

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Pokemon Gold/Silver from the original Red/Blue. Even if those are also somewhat dated by today's standards, the jump between Gen 1 and Gen 2 is the biggest the series did, even after all this years.