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
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.
Running a swf file locally doesn't really work for applications that have been pushed out to end users. The user's won't have access to any updates of the application (which an Air application does). Also a swf run locally might not be able to connect back to the server it came from, depending on the security settings.
In reality the nice part of flex is you don't have to commit to web or air. You can develop an application that can be built to run in both the web and on the desktop. For example you can put a majority of the code in a library project and then have a web specific and desktop specific projects that use that library.
Two other advantages of Air - one is that it allows you manage updates to your application and have the updates automatically pushed out. Second is it supports drag and drop (similar to native file access) so you can support dragging pictures or files onto your application.
AIR apps run using Adobe AIR runtime. It has access to your file system and includes more powerful capabilities. Whereas, a web-based flash/flex app runs in a flash player within a web browser. It runs in a sandbox. It has limitations. It can't access local file systems (without user intervention). Features of AIR: Can access file system. With AIR 2.0 you can access (communicate) to native file (executables) of that OS. Means you can communicate to a C app which does some OS specific functionality from the AIR app. Encrypted Local Store for securely storing credentials. Easy drag-and-drop functionality. Behaves as native desktop applications. Notifications. System tray icons. .. and much more.