I have both. Size A Aeron at home and a Steelcase Leap V2 at work. They are more or less base models so thats what I'm comparing and that is what you will find most often on the secondary market.
If you buy them new you can trick them out with ridiculous extras and spend thousands (on top of the base model price). For example, although I think the build quality of the base Leap V2 is worse than the base Aeron, you can customise a Leap with a polished steel frame and leather padding. So my criticisms of its fit and finish don't apply if you are prepared to throw eye watering amounts of money at it.
Aeron:
Comes in 3 sizes, A, B and C and you need to get the right size for your height, weight and body shape. Like clothing, you can find yourself between sizes or you can buy ill fitting clothes, in which case you will have a bad time. So its not a chair that works for everyone.
The plastic rim of the seat pan and back pan can cause problems. If your legs are too short (like mine), the rock hard plastic digs into the back of your knees and you get pins and needles. Or you stretch and your shoulder blades catch on the rock hard plastic rim of the back pan.
It is not a very adjustable chair. It has 2 backstops and I don't count the forward tilt lock as the third stop because its such an odd position to sit in. Sitting in forward tilt locked position is like being in danger of permanently falling out of your chair. I don't get it.
Aeron's seat pan doesn't move, the arms go up and down and can be tilted out but not very far. Aeron is a better built chair than Leap V2 in terms of fit and finish imo. More of the frame is metal and there is not a lot of movement/play in the adjustable parts. It feels solid.
I much prefer the Aeron tilt mechanism to Leap. Aeron's tilt is like some spring tension mechanism and its smooth all the way back.
You can get some rocking chair action going on with the Aeron. I leave it on the 2nd stop the entire time and just firm up the tilt until I lean back and kinda just hang in that position under my own body weight, without hitting the stop. Then I rock back and forth gently. If you like rocking chairs, Aeron is waaay more of one than Leap. Leap's seat pan slides forward when you lean back so you can't rock on it.
You can't sit in an Aeron with your legs crossed or in weird body pretzel positions. Well, you can but you will have a bad time. There is one way to sit in a correctly sized Aeron and all other ways are just varying degrees of weird and uncomfortable. But that one way feels pretty great to me and my persistent lumbar and sacrum pain is mostly gone.
If you are between sizes (like size A is too small and size B is too big), you may never have a good time. So just like clothes. I'm 5"5, weigh 65kg and Aeron size A is a decent fit for me.
Leap V2:
This is a "one size fits all" chair which means its highly adjustable to fit lots of different body sizes/shapes. The seat pan moves backwards and forwards. The lower and upper back firmness can be adjusted by tightening the slats. There are 5 backstops. The 4D arms go up and down and side to side.
Like one sized fits all clothing, you may find that because its made to fit a wide range of body shapes, it never quite fits your body shape either so you may end up flipping paddles and changing your sitting position and none of them might feel particularly terrible but none of them feel like a good fit either.
Leap V2 has a lot more plastic parts than the Aeron and the seat pan wobbles, the arms wobble etc. But to some extent you expect this because they are designed to move around.
I don't think Leap V2 is a comfortable chair or is even designed to be comfortable. I think its more designed so that your body doesn't collapse into positions that Steelcase's research suggests will give rise to common types of occupational injury.
The padding is minimal, especially on the back pan and its basically like dish soap sponge. It compresses flat under your weight and then it feels like you are sitting on a bench. The buzz fabric (the stock one) is very rough.
You may look at the padding in pictures and think its a plush chair but its not. Its much closer to a hard chair than a soft chair.
If you set it up right, its kind of impossible to slouch. As in, its very uncomfortable to do so. So I end up readjusting it throughout the day, flipping through different backstops and changing the back firmness, standing up, walking around, stretching and sitting back down a lot.
Leap's tilt/recline is some friction based mechanism. When you lean back there is a point where your weight overcomes some coefficient of friction between these plastic sliding bits and then you can feel it judder before it reclines smoothly. I don't particularly like the way it reclines.
Leap doesn't have the problems Aeron has with its plastic rims because the backpan is all flexible plastic so it moves around when you body does (less so if you firm up the back tension). The seat pan can be shunted back for short legs like mine.
This chair can't rock because the seat pan slides forward when you recline back. Some people really like this however.
Lumbar supports:
The lumbar supports for both chairs suck and I took them out. It just feels like someone is sticking their elbow into your lower back all the time.
Aeron's lumbar pad is a weird shape on both sides and too pronounced for my liking. Leap V2 lumbar support is a strip of plastic and to be blunt, it feels like it but you can control how much it digs into your back more with the lower back firmness dial.
I don't like either of them digging in my back so I took them out. Aeron has an optional attachment called a posture fit. There is an old and a new version. The old one is like a sacrum support (tailbone). I've never used it so can't comment. The new posture fit is like a combo lumbar pad and sacrum support. Haven't tried that either.
Shoulder/head/neck support:
If your sitting problems are related to upper back, shoulders and neck, neither of these chairs will help you. They are both low backed chairs. There is no support above your arm pits. The headrest attachments you can buy for both chairs are generally bad imo, although Steelcase's head rest attachment is better than the third party headrests for the Aeron (like Atlas).
I bought an Atlas headrest and ended up returning it because it doesn't go high enough for me. I hated the angle and it basically just forced my head forward and down so I was forced into slumping my shoulders.
Both of them suck imo and both chairs were never designed for this type of support so they end up just being an afterthought bolt on extra. Your mileage will vary depending on height/body shape. Neither of them worked for me.
Dietary considerations:
If you fart on an Aeron, everyone will hear it. The mesh doesn't exactly have spectacular sound damping properties, or really any sound damping at all. If you fart on a Leap V2, you can get away with silent killers since the foam muffles it. Your co-workers will know someone dropped a bomb, but they will never know it was you.
Wear and aftermarket parts:
Both chairs are pretty hard wearing. I suppose they were designed to extract the most productivity out of professional desk jockeys. Office workers tend to treat company property badly compared to stuff we buy with our own money, so both chairs can take serious punishment.
Aeron's frame especially is so rock hard that if you wheel it around and bang into furniture, its going to do much more damage to your cabinets and tables than the other way around. Seriously, if you have nice, hand carved rosewood cabinets, be careful.
If you need spare parts for self repair/upgrade, it is much cheaper for Aerons. Aeron has a peculiar and unique history - it was THE dot.com crash chair. Tech/web startups the world over bought dozens of them for their boardrooms in the late 90s. Amazon won, everyone else lost and all the furniture from the losers in the early 2000s ended up flooding the secondary market via liquidator auctions. Spare parts from unofficial sources were relatively plentiful for a long time, mainly from cannibalised Aerons that sat in warehouses for too long and were too expensive to refurb into a shape that is re-sellable.
You will still pay like 50 quid for a foam insert so don't get the impression Aeron spares are cheap by any means but oh boy do Steelcase fleece you aftermarket, if you bought second hand (out of warranty). If you buy new (with 10 or 12 year warranty or whatever), you can get any spares you need for free, which is nice.
Leaps didn't flood the secondary market like Aerons did and Steelcase has an iron grip on aftermarket parts. So you can only really get them via authorised distributors. The quote I got for a replacement pair of 4D arms was 258 quid, ex VAT and delivery. So yeah, I passed on that. Its a similar story if you lose your crappy plastic lumbar strip and want to buy a replacement. You will pay out the nose for it.
Conclusions:
Overall I prefer Aeron but mainly because I got lucky and it fits me better. I find it difficult to get a good fit in the Leap V2 and fidget/play with it a lot. Leap V2 is quite a big chair. However if you find the Leap V2 fits you better, its not hard to see why people prefer it to Aeron, but they are very different chairs. The main thing is you get a chair that fits you, but the best way to find out if it does is to go find a place locally that sells one and sit in it.