Let me analyze them, considering library with both exclusives and multiplats, controller with the ones that came with the systems, and player experience with the system UI, hardware and performance. Without considering graphics as asked. Giving points from 1 to 10 to see the difference. The points are based on how I perceive them nowdays, not only when they were current gen.

There won't be one for Microsoft as I've basically only played the 360 up until now, so I can't compare.

Nintendo

I've never played an actual NES, but have played its clone systems and its games, regarding the experience I'll extrapole on it from that plus all the videos I've seem.

NES to SNES (+6). Library: 6 to 8 (2). Controller: 3 to 6 (3). Experience: 4 to 5 (1).

SNES to N64 (-3). Library: 8 to 6 (-2). Controller: 6 to 5 (-1). Experience: 5 to 5 (0).

N64 to GC (+3) . Library: 6 to 6 (0). Controller: 5 to 7 (2). Experience: 5 to 6 (1).

GC to NW (-6). Library: 6 to 4 (-2). Controller: 7 to 4 (-3). Experience: 6 to 5 (-1).

NW to NWU (+5). Library: 4 to 6 (2). Controller: 4 to 6 (2). Experience: 5 to 6 (1).

Even as the Switch 1 and 2 are handhelds, they succeeded the Wii U library, so I'll continue to them.

NWU to NS1 (+1). Library: 6 to 7 (1). Controller: 6 to 6 (0). Experience: 6 to 6 (0).

I'll consider the Switch 2 library for its potential, as it's too new.

NS1 to NS2 (+3) . Library: 7 to 8 (1). Controller: 6 to 7 (1). Experience: 6 to 7 (1).

Overall.

NES 6/3/4
SNES 8/6/5
N64 6/5/5
GC 6/7/6
NW 4/4/5
NWU 6/6/6
NS1 7/6/6
NS2 8/7/7

Sega

I can only make 1 comparison as I've never played the Saturn or the Dreamcast.

SMS to SMD (+7). Library: 4 to 7 (3). Controller: 3 to 6 (3). Experience: 4 to 5 (1)

Overall

SMS 4/3/4
SMD 7/6/5

Sony

PS1 to PS2 (-1) . Library: 10 to 9 (-1). Controller: 7 to 7 (0). Experience: 6 to 6 (0).

PS2 to PS3 (0). Library: 9 to 8 (-1). Controller: 7 to 7 (0). Experience: 6 to 7 (1).

PS3 to PS4 (+4). Library: 8 to 9 (1). Controller: 7 to 9 (2). Experience: 7 to 8 (1).

PS4 to PS5 (+3). Library: 9 to 9 (0). Controller 9 to 10 (1). Experience: 8 to 10 (2).

Overall

PS1 10/7/6
PS2 9/7/6
PS3 8/7/7
PS4 9/9/8
PS5 9/10/10

Conclusion: based on crude points only, the Sega Mega Drive would be the biggest improvenent, with the SNES and Wii U following.

Even so, my pick will still be the PS5.

It makes me literally don't want to go back to playing on other system with how good it is, as you can see in the overall scores.

Games load nearly instantly, the majority of them offer 60fps, the UI is fast, it has the best controller I've ever used, and despite online criticism it has a lot ot great games, and pretty much all of the current gen games.

Other system had more room to improve upon, but despite the PS4 being so good already the PS5 still managed to improve so much on something already so good that it makes it the nobrainer choice. If only the PS5 had physical BC to all other PS gens...

SNES obviously, that's part of the reason it's so beloved. Switch might be second place, that's definitely also a huge leap.

Yup. Wii U has some gems and has more personality in its OS, but is slower, clunkier and with less games worth playing.

If you're not a collector or modder, a Wii U has been virtually worthless since around 2018-2019. It can play Wii games, sure. But just get a Wii then. 

Visuals can be a factor I suppose, I just wanted to be clear that the question isn't "biggest graphical leap".

Wii U to Switch, not sure it has to be explained because the Wii U was a flaming pile of hot garbage.

Wii U Library was actually descent. Their mistake was the tablet controller. They should ditch the tablet and sell it for 100$ less.
It would have sold 20-30 millions easy like Gamecube/Nintendo 64.

PS1->PS2

From warpy, jittery, low-fps to stable, often 60fps 480p graphics. Modern control schemes with dual analog became standard. DVD player for free as well.

Here are the ones that stick out in my head (going from least notable to most notable):

5) PS1 to PS2 - I put this in 5th place, because there are a lot of great games on the PS1, but to me it is mostly an RPG system.  The PS1 shined when it had huge games that were slow like FF7.  Then the PS2 comes along and load times aren't an issue anymore.  The PS2 has great RPG's also, but it also has fun action games.  PS2 just has a more diverse library of fun titles compared to the PS1.

4) SMS to Genesis - SMS was a technically impressive system that was just getting steamrolled by the NES.  The NES was simply creating several of the most iconic game series of all time and it also got all of the third party games.  Then the Genesis comes along.  It has 3 Sonic games not even counting the CD-Rom or 32X, while the SNES only has 1 Mario game.  Genesis also got a very good third party library of games, while Sega was putting out some of the best games it ever made like the Shining Force and Streets of Rage games and also arcade ports like Golden Axe.  The Genesis is the reason why Sega is still thought of so fondly today by many people.

3) Wii U to Switch - The Wii U was a flop with a few fun games.  The Switch comes along and has the biggest game library ever.  Saying there is something for everybody on the Switch is an understatement.  Combining the home and handheld resources really did lead to a ton of games, although most of them are actually third party games.  And of course the fact that it can be played at home or portably is the icing on the cake.

2) Gamecube to Wii - Gamecube games are not fun.  It was just an inferior iteration of the N64, including failing even harder than the N64.  Then comes the Wii as the most innovative system ever made.  Wii games are pure fun, and they are like nothing else.  The Wii also had a library several times larger than the Gamecube, and it brought Nintendo back to the perception of being a serious developer in the home console space.

1) Gameboy to GBA - This assumes you count Gameboy to GBA as one generational step.  But so much happened from the original Gameboy to the GBA SP.  Consider that the original Gameboy was a black-and-white brick that ran on batteries and was not backlit.  Gradually all of that changed, first color, then some shoulder buttons, but the SP model really innovated with chargeable batteries, a backlit screen and a clamshell design.  On top of that it has this fantastic 2D library of games including many NES and SNES ports, as well as a huge library of original games.  There are even some PS1 style games like FF Tactics and 2D Castlevania.  Handheld gaming really came into it's own with the GBA SP model, and there was this great, diverse library of 2D games to go with it.  For me this has to be the winner of biggest improvement over its predecessor.

NES to SNES. A Link to the Past and Super Metroid alone > LoZ and Metroid, and Square's RPGs. 

N64 to Gamecube. The graphics and controller aside, the Gamecube had a game library I enjoyed much more than the N64 library. I enjoyed TP and WW more than OoT and MM, I'd rather play RE4 or Metroid Prime than Goldeneye, and the Gamecube had some actual honest-to-god RPGs, and a couple of solid fighting games.

Wii U to Switch, as everybody else has said. With that said, I'm really enjoying my Switch 2.

PS1 to PS2 is kind of weird for me. The PS2 was a great system, no doubt, but the PS1 had a lot of RPGs that I loved, and the PS2 didn't have quite as many. I was also playing the Gamecube a lot more in relation to PS2 than I was with the N64 versus the PS1. 

Wii U to Switch or 3DS to Switch for that matter sounds so wrong to me. While we certainly got quite the upgrade in terms of catalogue, we lost SO MUCH in the process:

- Free online
- Virtual Console
- 3D without glasses
- Actual portable devices that could fit on your pockets
- The ability to browse the internet and/or perform side tasks (write notes for example) while gaming on the same device
- Dual Screen gaming (losing unique gameplay conepts and styles, something very aparent with Wii U/DS/3DS ports/remakes for Switch/Switch 2)
- Accurate pointer controls
- Related to the previous two points: inferior ports/remakes that presents themselves as "Deluxe" versions or outright replacements
- Lesser scope games that would come to traditional Nintendo handhelds for 39.99-44.99 suddenly sold at 59.99 just because "the Switch is a home console now".

And probably many more that I'm forgetting about right now. So no, I don't see the Switch as "one of the biggest improvements over its predecessor". In many aspects, quite the contrary actually (and I still love the Switch/Switch 2 mind you).