I've recently gone from a JCW hatch to an S3 S-tronic (albeit with a short spell in a Mk1 TT in between).
I've just picked up my S3, so probably still in the honeymoon period, but I will try to be as objective as possible, they are very different cars so quite hard to compare.
My JCW was pretty well specified, I had the dynamico recaro's, harmon kardon, nav, mini connected etc, pretty much every option ticked ..
I love the way it looked, and it always gave me a smile when I saw it parked up. The exhaust pops and bangs are addictive, and I think that they've judged sport mode well, there's a noticeable difference in throttle response in sport (something I'm not quite feeling in the S3 yet with all the different modes). It's a torquey wee motor, and in sport mode felt very responsive although I think it felt a bit breathless higher up the rev range. It felt quicker than it was in reality I think. I had sports suspension too, but I felt this was a bit hard for our roads and you would get bounced around a bit too much. Interior quality was fine, some of the plastics felt a bit cheap and I wasn't keen on the wee joystick. Loved the alcantara seats and steering wheel. It's basically a wee go kart, epic fun when you're in the mood, but not so good when you're in a 'I just want to relax' and get there mood.
To try and compare it to the S3, well it's bigger that's for sure. It feels a step up in quality, from the interior to the way it insulates you a bit from noise/vibration/harshness. I've gone S-tronic, so can't compare gearchanges, but loving the S-tronic so far. I don't miss the pops and bangs from the Mini, as the exhaust burp more than makes up for it, and the S3 has a really nice brrr as you rev it out. With almost 300 bhp/300 lb/ft it feels a much stonger (and smoother) engine and feel happier revving out to the redline. I think the S-tronic helps here as you're not worried about banging it off the limiter.
For me, I think the S3 can be 2 cars. Very relaxing around town with the S-tronic left to do it's thing, and also long distance where it's quiet and does the motorway cruise in comfort. When you want a blast though, get it set to dynamic, move the gearlever across to manual mode it really comes alive, it's pretty epic.
Will you have more fun in the Mini? Hmmm, probably, it's just such a cool wee car and the Audi will not give you that go cart feel. You do feel a bit more removed in the Audi (but will likely be far quicker).
If you had to choose one car, on a sunny day for a wee B-road wiggle, and you're not caring about how quick you're going then the Mini would have the edge for fun.
However I don't think the S3 would be far behind, and for all other times it's the easier car to live with.
Not sure that's helped you much, I know how JCW's can get under your skin.