What is Flex
Flex is a powerful, open source application framework that allows you to easily build mobile applications for iOS, Android™, and BlackBerry Tablet OS devices, as well as traditional applications for browser and desktop application.
Flex has different SDK Version, more details
The Flex SDK contains hundreds of out-of-the-box components. DataGrids, Charts, Formatters, Validators, and numerous other UI controls are the building blocks for applications of all sizes. Components can be styled and skinned to fit the look and feel you want. There are also hundreds of third party open source and commercial components available for Flex. Mapping APIs, Data Visualization libraries, and Cloud APIs provide the building blocks for assembling great applications.
You will use the following two languages two develop flex applications.
1 . MXML is an XML-based markup language that is primarily used to layout application display elements.
2 . ActionScript is an ECMAScript-compliant object-oriented programming language that is primarily used for application logic
What is Flash Builder
Flash Builder is an enterprise-class Eclipse-based IDE to develop Flex applications.
Flash Builder includes state-of-the-art productivity features (code assist, refactoring, etc),
on-device debugging, and mobile simulators that allow developers to test their applications
on different screen sizes and densities. Flash Builder allows developers to package native
application files (ipa, apk, bar) for release on the Apple App Store, Android Marketplace,
and BlackBerry App world.
What is AIR
Adobe AIR is a cross-operating system runtime that enables you to use your existing ActionScript or HTML/JavaScript development skills and tools to build and deploy rich Internet applications (RIAs) outside the browser(Desktop) and on mobile devices.
Best Answer
Flex has much rich UI experience, provides more functionalities, requires too less a coding, is compiled at server side and pushed to client in binary form (as SWF file), is client independent (so you don't have to make browser tweaks) and has pretty much tighter integrations with server side technologies. Integrated with LiveCycle (LCDS) you could have server push data to flex clients, rather than pull based mechanism. The list of advantages would go on and on...