Alright, I\'m confused by all the buzzwords and press release bingo going on.
Flex runs on Linux, too, while Flash doesn't.
Flex is kinda Flash CS 4 second edition.
Flex is less graphical, as it separates compiler and IDE, which allows for command line compilation (makefiles, large projects so to say) which allows for alternative IDE's to Flash.
Edit:
Flex lacks some classes that Flash CSX has (e.g. fl.controls), while Flash lacks some classes that Flex has (e.g. mx.controls or mx.alert).
All in all:
You can have your own Flash compiler for free by downloading Flex 4 SDK and FlashDevelop.
But it is no substitute for Flash.
Flash produces much smaller files (e.g. Flash compiles a project to 100 kB while Flex is compiles the same project to 500kB). So Flash is for internet multimedia applications, while Flex is for desktop multimedia applications.