I'm a longtime Model S owner. What
@glide says is correct, the Y back seat is much more comfortable for 1 or 2 adult-sized people than the S. The Y seat is higher off the floor, and the Y has more headroom. However I believe the S is wider, so for 3 across the S might be better, especially for kids if they're not yet adult height (i.e. if the Y's extra seat height doesn't matter yet for them).
@glide is also correct about Y ride quality being significantly worse than any S with air suspension. The Y ride is very busy, you feel everything even over fairly smooth (but not perfectly smooth) roads. Some people don't mind it at all, while others (like me) find it borderline intolerable. To me it feels very out of character for a modern family CUV.
From what I've read, some of the higher quality aftermarket damper+spring setups might do a lot to improve ride quality (while also increasing handling capability at the same time), however you may or may not be comfortable modifying the suspension on your family car. (Don't bother just changing out springs, the dampers are the biggest part of the problem as best I can tell.) I ordered a set of Redwood Motorsports Öhlins DFV coilovers (dampers+springs) for my 2021 Model 3 Performance before I even picked up the car. (I did this based on my test drives, including a non-Tesla EV that came with Öhlins DFV dampers from the factory.) They have a Model Y version. If I ever buy a Model Y (definitely a possibility if I need to replace my old S) I will buy the Öhlins for it. On my Model 3 I went for "performance sport" however on a Y I would just go for "grand touring" (softer).
Other stuff...
I hate the 3/Y turn stalk. Hate it. So Much. It always returns to center, so you can't feel where it's at, and half the time I try to turn off a turn signal, it either doesn't respond or it switches the signal to the opposite side.
I've barely driven my 3 in the rain and I already miss having full wiper controls on the stalk. This is partly because the autowipers are worse on my 2021 Model 3 than my 2013 Model S. I didn't test drive the Y in the rain but I'm pretty sure its autowipers work the same as the 3. There is a button for getting 1 wipe (long press to spray washer fluid) that also directly brings up the touchscreen wiper settings, which are positioned on the driver's side (at least for left-hand drive), so that helps somewhat!
Forward(+side) visibility is hugely better in the Y than the S. The beltline feels very low relative to your seating, the glass is tall, it feels almost old school car and way better than most modern cars. I love this about the Y. So does my wife.
Coming from an S, the 3/Y center display map feels small. The display itself is a little smaller but that's not the issue. The problem is Tesla took up
way too much of the center display on graphics that were in the driver's display. The layout could be much more space efficient, a huge amount of space is wasted displaying car graphics and the map cannot be extended over it.
My wife really misses having a roof that opens. (Though not all S have that, not sure if yours did.)
The 2021 Model Y I tested had loud wind noise on the highway (by modern car standards), worse than my old 2013 S, let alone a newer (quieter) S. However I don't see too many Y wind noise complaints on these forums, so I'm suspicious that my test car ran afoul of Tesla's legendary build consistency. If that's the case, Tesla service
should be able to fix this if your car is affected.
I like the front seats of an S of your era more than 3/Y front seats. This is personal preference to some extent though. I do like the 3/Y front seats more than my 2013 S front seats.
The 3/Y steering ratio is much quicker than S steering ratio, and the 3/Y steering wheel is smaller too, accentuating the ratio difference. In the sporty-feeling Model 3 I love this. In the Y it feels almost too fast, a bit out of character. I might prefer a
slightly slower ratio in the Y. I'm sure you'll get used to it though, this is just an initial adjustment thing, and better the steering is a little quick than too slow!
The dual motor Y frunk is much smaller than my RWD S frunk. However since you're coming from a dual motor S, I think the Y frunk is comparable.