The devices make different trade-offs and so appeal to different people. Personally, I prefer the mini, but they are both great tablets. While I haven't owned the air, I'm going to give my thoughts anyway.
Advantages of the Air:
In my experience, browsing web pages is often more natural on a larger screen; on the mini I often find myself enlarging text or switching to portrait view on. The onscreen buttons, tabs, and controls can be a little harder to press on the mini; they sometimes feel as though they were designed for a larger screen and then shrunk down. I also think watching movies is more enjoyable on a larger device, and rarely use my mini for that. And while the Air is larger than the mini, it's by no means huge. Light and about as small as it can be given it's screen, it's easily carried or slipped into a briefcase or backpack or even slipped under an arm. I was never interested in the original Ipads, but the Air is small and light enough I might have gotten it if the mini weren't available.
Advantages of the mini:
The mini slips easily into a typical jacket pocket. As I wear a jacket anyway, it's always there when I need it. I don't even need to reach for a briefcase or bag, it's always right near my left hand, ready to be pulled out in less than a second whenever I have a need for it. And it never gets in the way. I'm an instructor, and typically use it for storing my textbooks and worksheets, and so when lecturing at the board I'll hold it one hand while I use the other to copy down problems with chalk. I'll often have it out when navigating an unfamiliar neighborhood or even a mall (Google maps actually had a map of my local two story mall). If I'm in a crowded take out place, I can stand in a corner and read, without it getting in the way. And of course when I no longer need it and its time to pick up my order, no need to reach for my briefcase to put it away, no need to even have my briefcase, it just gets tucked right back into the jacket pocket so I can pick up my order with both hands and leave. I also personally find the mini to be a more comfortable size for reading. When reading, I prefer to be able to get an overview of the page I'm looking at, and the Air screen feels to big for that; my eyes have to move around more than I would prefer.
A previous thread compared the Air to a hardback book and the mini to a paperback, and that seemed an apt comparison to me, they both have their advantages. People who prefer the Air will sometimes say if I want something portable, I'd just assume use my phone for that. People who prefer the Mini will often say, if I want something larger, I'd just assume use my laptop for that. That's how I feel, as small and light as the Air is, I'd need some sort of bag of some sort to carry it most the time, and once I have a bag, I might as well have a laptop. That's the reason Ipads didn't appeal to me before the mini came out.
But they're both fine devices with different strengths that naturally appeal to different people; I don't think one could sensibly say one of them was somehow objectively better overall than the other.
