Apache Flex (formerly Adobe Flex) is an open source application framework for building and maintaining expressive web applications that deploy consistently on all major browsers, desktops, and devices. It provides a modern, standards-based language and programming model that supports common design patterns suitable for developers from many backgrounds.
Adobe AIR is well-suited for a developer who is already familiar with Flash and AS3. It is well-suited for 2D app development using Starling or another third-party graphics library. However, to be used appropriately and avoid errors and poor performance, a basic understanding of programming principles is needed, so it's not great for a designer who hasn't studied a lot of coding.
I feel like it was better suited in its role in the early days of the app, as things progressed, it didn't as much. It may have gotten better as Apache took over, but I think they lost some users due to other things taking over on the web a bit more.
Although Adobe AIR is just an SDK without an actual "UI" it's commonly used within Flash, Flash Builder, or FlashDevelop. Considering the integration with Flash IDE, there are very few tools that can compete with its features.
There are definitely other platforms out there that are awesome but Adobe AIR is just what our other developers were used to using but we do have some success.
Apache Flex is a modern, standards-based programming language that supports common design patterns. It's an open-source SDK used to make rich Internet applications that work on different browsers and platforms. Any flash-based application framework, such as Eclipse or the Flash above Builder, can be used to implement this. So far, in comparison, it's much more crucial, especially for people working in fields like mobile development.
Using Adobe AIR allows your team to be comprised of Flash/ActionScript 3 developers, which can be easy/cheap to find compared to other disciplines (based on location). This can lower the bottom-line in terms of engineering costs.
As the Adobe AIR SDK is free to use, the cost of entry is basically free. Additionally, the Adobe AIR SDK may be used in conjunction with a free IDE, such as FlashDevelop, making the entire stack completely free to use.
One negative impact is the stigma against Adobe Flash/AIR. Due to Adobe's less than stellar track record (in terms of support and security) with both Flash and Flash Mobile, there are numerous clients that we've dealt with, that refused to approve Adobe AIR as a tool.