Please feel free to add multiple answers, each with a single point, to make the voting work well… and don’t add something someone else already said.
All I know about AIR is you can have local storage, but there must be more to it… offline access is another key one but then you can run a SWF locally if you save it to your PC.
The biggest benefits (in addition to LocalStorage/SqlLite support) would be…
In addition, Air apps aren’t just Flash based, but can also use HTML/JavaScript via WebKit
Update:
All the info you need on Air is on the Wikipedia page. (and it’s external links) http://en.wikipedia.org/wiki/Adobe_Integrated_Runtime
In short, if you need native filesystem and limited desktop interaction Air is the way to go, if you don’t need these features, use Flex in the browser.
If you need additional system integration but still cross platform, try Appcelerator/Titanium, Java/JavaFX, Python with WxWidgets, or C++/QT.
It really depends on what you want to build.