We get this question multiple times a week. I think I need to write up a response and then copy/paste.
The bottom line is both devices have their advantages and their ardent fans. Which one is best is mostly a subjective decision, and only you can decide which is right for you.
Having said that, I'm a big fan of the mini; for me it's the perfect form factor. It's just barely small enough to slip into a suit jacket pocket and go everywhere with me, and just large enough to comfortably display pdf files which aren't easily reformatted for a smaller screen (again that's for me; a few have complained the smaller print is hard to read). My mini goes everywhere with me, and never gets in the way, even when I'm holding it one hand to use GPS in a shopping mall with a bag in my other, or when I'm reading problems out of it while writing them on the chalkboard for my students, or when I'm reading it standing in a line somewhere, or using it as a GPS in my car. Reading books feels a little more natural to me on the more modest screen. Some have compared the mini to a paperback book and the Air to hardback.
On the other hand the Air is a beautiful machine, and may be the right choice for most. The OS was originally designed with the Air in mind, and buttons and controls can be a little finicky to press on a smaller device. To me, movies look nicer on a bigger display, and text can be a little less cramped. While not as small as the mini, it's impressively sleek and svelte. For me, the incredible portability of the mini trumps the advantages of the Air, but for others the reverse is going to be the case.