Firstly: I’ve used film, full frame, MFT, and APS-C cameras over the past few decades.
The summary of this very long post: Sensor size affects lens size and weight. Don’t let size and weight stop you from bringing a camera.
All the details, with photo comparisons and everything…
Full frame (and film) is really only “better” at a few niche things at this point in time:
- Extreme background blur (but really not that much more than APS-C, when you have the right lenses — I often have to stop down lenses like my Fuji 35 f/1.4 and 56 f/1.2
You can get absurd blur in APS-C and there are even “cheap” f/0.95 lenses for even more if you want it). - Adapting antique lenses (full frame will use the lenses fully… and APS-C will either only use the “sweet spot” or require a “focal reducer” adapter, aka: “speed booster”)
- Wide angle (but there are excellent ultrawides these days, even on APS-C, and going extremely wide, as mentioned above, has challenges — I just got a Sigma 10 - 18 f/2.8 for my Fuji and it’s superwide… more than close enough to when I used my Sigma 12 - 24 on my 5DmkII)
Most of these are irrelevant in most cases, except the adapted lenses (unless you get a good speedbooster — skip the cheaper Zhongyi Lens Turbo and go with something like a Metabones or a Viltrox — or are fine with the crop).
A long while ago, it also used to be the case that you’d need full frame for very high megapixel images (without stitching), but that changed over the years. (The X-T5 is 40 megapixels, which is an excessive amount for most photos, but means we can crop a lot.) Low light and dynamic range were also reasons for a time, but APS-C and even MFT caught up as sensor tech got better. Everything is really absurd in low light now, especially if you have an ISO invariant camera (not all are, but I know Fujis and Ricohs are), which can pull up shadows to an impressive degree.
However, something that every full frame brings along with it versus APS-C and MFT: Larger, heavier lenses overall, especially for zooms and higher quality lenses. If you’re shooting with primes, you can get away with smaller and lighter lenses sometimes, but you could find even smaller and lighter lenses on APS-C and MFT too. The more lenses you haul around and the longer you haul them around, the more fatigued and annoyed you’ll be, until you decide to not bring them along all that much.
Here’s a an extremely quick and dirty comparison of my new Sigma 10 - 18 f/2.8 (constant aperture) on my X-T5 versus my old Sigma 12 - 24 f/4.5 - f/5.6.
That’s visibly a big difference in size and the weight is 833 vs. 1425 (the Fuji is 0.58 the weight of the Canon). Apologies for the quick and cruddy photo from my Pixel 6 Pro. It’s night here (so it’s quite dark), and I just have two banged up scales in the kitchen.
The thing is, this is not exactly a fair comparison:
- Fuji and the new Sigma lens are both much higher quality in every single way.
- The newer camera and lens are also a decade and a half newer.
- The Sigma lens, despite being smaller and lighter and higher quality overall, also has a constant wider aperture! It’s some magical feat. I’d imagine if they had a higher f/stop, Sigma could probably figure out a way to make it even smaller and lighter somehow. (FWIW: There’s the Laowa 9mm manual prime for APS-C that’s even smaller and a touch wider.)
- In addition to the aperture difference, the new Sigma focus much, much, much closer… so if you want a ridiculous photo that’s wide angle and has a lot of background blur with good bokeh, this can deliver. The full frame lens can’t. (Background blur and bokeh is usually odd on an ultrawide, really.)
- The range is pretty close, but not exact. The old Sigma on the full frame Canon can go a little wider, and the new Sigma can go a little closer. It’s a 15 - 27 equivalent for the new lens vs. 12 - 24 on full frame. But they’re both pretty close, really. The old Sigma gets very distorted with low quality and vignetting when zoomed out fully wide on a full frame camera, however, so I often had to zoom in a little.
If it’s a still subject and you don’t shoot wide often, consider shooting multiple images and using a stitching program XPano or Hugin instead. 
Another fun “equivalent” comparison:
(I should’ve kept the paper underneath to make the cameras more obvious against the backdrop. Ah well.)
The Canon is considerably larger than the Fuji. The size of lenses goes generally goes up more the more you want to zoom. The weight is 895 vs. 1528. Here, the Fuji is 0.585 the weight of the Canon. (Yet again, almost half.)
This is the kit lens of each system, and they’re close to equivalent lengths for most purposes. Both lenses are considered very, very good; people have said “the quality of a prime, but it zooms” for each.
- The Fuji is still smaller and lighter.
- Fuji is 18 - 55 f/2.8 - 4, which is the equivalent to 27 - 82.5; Canon is 24 - 105 f/4 (constant).
- The Fuji lens lets in more light when it isn’t zoomed in. When fully zoomed in, they’re both f/4, and the subject isolation will be more on the full frame Canon for these two lenses (due to sensor sizes).
Okay, now let’s kick it into ludicrous size:
The Canon is absurdly larger and even more weighty. It weight so much that the old scale (which is yellowed) had “Err” on the screen, so I had to swap the side for each camera… 
This is the Canon 70 - 200 f/2.8 L IS vs. the Fuji 70 - 300 f/4 - f/5.6. The Canon is 2333 vs 1160 for the Fuji. This time, the Fuji is 0.497 the weight of the Canon.
- The Fuji is higher quality and zooms more (especially on an APS-C, so it’s 300 x 1.5 versus 200)
- But the Canon lets in more light @ f/2.8. (This Canon zoom was top of the line when it came out in August of 2001. There were also variants without image stabilization and some at f/4. The f/4 versions were half as heavy.)
However, each of these examples are just a camera plus a lens. Now think about all the other lenses (1, 2, or more) you might be bringing along too. It all adds up quite a bit, for both space and weight, and the weight gets more annoying over time.
Everything above is “unfair” in one way or another. I’m comparing very old gear versus pretty new gear. There have been lots of advancements. All the mirrorless cameras these days are usually smaller and often lighter than a DSLR. However, a full frame camera isn’t going to be smaller or lighter than the Fuji — at best, it’ll be the same size and weight (it’s impressive what Sony has done, as they do have some approximately the same size and just ~120 grams heavier)… but the lenses will still be larger and heavier, especially for zooms. So the main point stands: If you upgrade the sensor size, you’re also increasing the size and weight of lenses, generally.
The thing is, if you get a couple of small primes and a reasonable zoom (I probably should’ve gone with the Canon F/4 back in the day ), you can have a small and light enough Sony or Nikon camera without all the size and weight. Primes are usually smaller and lighter than zooms. But not always. The full frame primes are still going to be larger and heavier overall, but the difference won’t be that big, as they’re smaller and lighter and simpler than zooms.
But: Please consider your back. It’s not fun when it hurts and you’re forced to change camera systems.
Extra bonus round: Let’s compare APS-C versus APS-C!
Here’s the Ricoh GRIIIx versus the Fuji X-T5. Both are APS-C. The Ricoh has a 26.1mm f/2.8 lens and the Fuji has the 27mm f/2.8 pancake. The Ricoh weighs 260 versus Fuji’s 645. The Ricoh is 0.4 times the Fuij.
Here’s the Ricoh versus my phone, the Pixel 6 Pro. My phone’s 245 grams… which is close to the weight of the Ricoh, really.
- The Ricoh fits in a pocket. (Not just a coat pocket, as the X-T5 can, but also even a jeans pocket.)
- The Pixel takes absolutely awful photos, as it’s a phone camera. It was considered “the best camera” for a phone when it came out in 2021. It’s awful though — the JPEGs are overbaked trash and the raw files are DNG and sometimes (but not always) pretty decent thanks to darktable. This is why I eventually bought a Ricoh GR, as I can’t always bring my Fuji along, but I can pretty much always bring the GR.
- It’s tiny. Much smaller than my MFT Panasonic GF1 + 20 f/1.7 pancake, which I gave to my sister. (The GF1 was around the same size as my Fuji + 27 f/2.8 pancake, and both are the same 40mm equiv.) Seriously, I don’t know how Ricoh shrunk down an APS-C camera with a great lens (that also has a macro mode) into something this small. Magic?
- This shows that any APS-C body maker can probably make their bodies at least a little smaller, but they might have to cut corners. Ricoh didn’t have to add in a mount, for example. Ricoh GR cameras also don’t have weather sealing (sadly). I do know that Fuji does have some smaller bodies than their X-T line (X-E4, for example); other APS-C makers might as well.
The point of this last one as well as the above comparison of full frame versus APS-C is this: Bring a good enough camera with you. If it’s smaller and lighter, you’re more likely to bring it with you. Switching to full frame means larger lenses, which will get in the way of you bringing your camera with you. (To varying degrees for everyone, but it larger and heavier will get in the way for everyone at some point in time.)
I personally think the APS-C sensor size, when you have lenses designed for it, is a perfect sweet spot when you consider everything. Others think that sweet spot is MFT. Some are happy enough with full frame. Others don’t mind the bulk of medium format somehow. Any modern enough camera will be great though, as long as you take it with you.
Don’t make a mistake of switching your camera body and lenses to have something that you realize is a chore to bring around however, especially as there isn’t much benefit for full frame versus APS-C, and there are drawbacks too (not just size and weight… if you want to zoom further, APS-C is actually better).