Some ideas/impressions I've got while comparing both.
In the hand
Although the main body of both the 12x36 III and 10x30 II (pictured) is actually the same piece, I find they feel surprisingly different in the hand. The 12x36 weights in at 660 g and the 10x30 at 600 g, so their "density" is really different. The 10x30 feels heavier, which (given the limited weight) is not necessarily a bad thing in this case: it feels more substantial, less plasticky. The 12x36 feels lighter, more plasticky and hence less robust. The 12x36 is thus clunkier. It's not a terribly large device, but it fills your hands. The 10x30 feels more zippy. Let's say the 10x30 feels more like an everyday pair while the 12x36 feels more (to me, this is) like a specialised tool.
Both Canon require you to constantly press the button whenever you want to make use of its IS function (to me it means 100 % of the time; that's why I use an IS device after all). Despite the body being the same, for some reason I find the 10x30 easier to use in this respect, I feel I'm less aware of the fact that I need to be pressing the button, with the 12x36 I feel I'm actively doing something, while with the 10x30 it feels as if the simple fact of grabbing the binoculars makes me inadvertently press the IS button.
Carrying them
This might sound daft, but to me how you can carry a pair of binoculars is a pretty relevant thing. For example, if I can fit them in my favourite case (from a 8x30 Monarch 7) it means I can comfortably carry them on my belt, which has the added benefit that I'll be grabbing them more often, even if I'm not going "serioiusly" birding, but just out for something like walking the dog.
The 10x30 pass this test (it is a snug fit, but it succeeds at it), which I really like, I find it so convenient, seriously. So, both in terms of "grabbability" and portability I can see the 10x30 IS II becoming my main (even only) pair. The only caveat is the rain (because neither of these Canon are waterproof), but then I live in the island of Ibiza, and we don't get a lot of rain here (maybe that's one of the reasons 42 mm binoculars don't appeal to me either; it is usually very bright here and dawn/twilight are brief periods where a x32 gives you basically 95 % of what you'll ever need in terms of brightness).
Given that I find the cases provided with both the 10x30 and 12x36 basically useless, I carry the 12x36 inside the case of an EL x32 or SLC x42 (it's the same). Noticeably larger than the other one, but at least it allows for a very comfortable "bandolier style" use while not using the binoculars. The strap (safety belt style) makes for a really comfy experience, even if you carry it for hours.
Eyecup comfort
I've read many criticism regarding the eyecups, and I agree that they're not the best parts of the binoculars. It must be said that, at least in my case, I can see the entire FOV without much problem (I don't wear spectacles). I would have preferred a more contemporary multiple-step twist-up design that allowed the viewer to achieve his or her perfect viewing position.
One funny thing: the right eyecup on my 12x36 IS III (the one controlling the dioptre setting) moves at the slightest input; for example, while taking them out of the case. It even moves during a single using session, which means some times I have to readjust. The one on the 10x30, on the other hand, stays in place without much problem. This is puzzling since both look identical, but one is clearly tighter.
And now for the optical bit (which was in fact the OP question)
The view is (as expected) very similar, given that an important part of both binoculars is basically the same. However, I'd say I feel the 10x30 give a somewhat "cleaner view". Comparing them side by side, I'd say the image through the 12x36 looks a little more washed, less contrasty (probably due to higher magnification). I'd also say that CA is a tad better on the 10x30, but not by a huge margin.
FOV feels nice on both. No, they're not class leaders, but I think 6º for a 10x and 5º for a 12x is more than adequate, I've never felt they suffered in this department. As a matter of fact, maybe it's because both IS are binoculars that require an extra step in your attention (constantly pressing the IS button), but I feel FOV is perfectly OK.
Under difficult light circumstances
This is one area where I find both struggle a bit. It can be usually improved with a slight tilt of the head to avoid the unwanted light veiling the view, sometimes fine-tuning the IPD also helps. So, not ideal for looking West during sunset.
Battery life
It's probably sample variation, but I find the batteries last way longer on the 12x36 IS III compared to the 10x30 IS II. I think this shouldn't probably be so, because both are the last iteration, but I get a noticeably larger battery life with the larger 12x36. This comes as a surprise, I would have assumed that, if one of the two was to be more efficient, it would have been the smaller 10x30. But who knows. Anyway, I get around 2,5 months of use with the 12x36 and less than 2 months (in fact, less than 1,5 months) with the 10x30, in regular use (using them around 3 - 5 times a week). This is just my personal experience with my two units. YMMV.
Conclusion
I feel the 10x30 is a more all-round device, is smaller, a little easier to use, more convenient to be carried, has a wider FOV, the dioptre setting stays in place, the image has a little more punch... However, the 12x stabilized image is really something to behold. No matter where I am, when birding with other people using non-IS 8x 10x, I'm usually the first to spot the defining detail, which to my eyes gives the 12x an invaluable advantage, thus making the most of the IS.
For this reason, if I had to have a single device to do it all, I'd take the 10x30, but if I had to choose one of these two to complement a non IS 8x32 (or 8x42), I'd definitely keep the 12x, even if I find it slightly "worse" (I'd better say "less convenient") in many areas. I think a compact 8x30-32 and a high power IS, like this 12x, is a perfect couple giving an incredible versatility and power. In fact, come to think about it, my beloved ELSV 8x32 have seen very little use ever since I bought my first IS in june 2021. Ever since, I basically always grab an IS unless I'm climbing very light (when I take the 8x32 Opticron Traveller) or if I know it's going to be raining a lot, when I do take the ELSV. Otherwise, it's an IS. If I'm going to anything but close forest, it's usually the 12x. If I'm going for some more "casual" birding (say, we're going on a hike or walking the dog with some casual birding possibilities, then it's the 10x30).