Flex is a library that sits on top of Flash that provides controls and a layout system.

Flash itself doesn't have the concept of a checkbox, drop down list, grids or any controls beyond a button. Flex adds these.

Flash also doesn't have any sort of layout system other than absolute positioning. Flex adds various layout controls.

It works pretty well, but also increases the size of your .swf file quite a bit.

Adding to the other comments, I find it easiest to think of Flex as a Flash-version of HTML, CSS, JavaScript, and pick-your-favorite-JS-libraries such as jQuery and Sencha. Instead of HTML, Flex has MXML. Instead of JS, it has ActionScript. (Though both come from the same language standard.) CSS works, and they bundle a good set of default themes and widgets. I think it's a really nice piece of work, especially when they put out Flex 4, it's a joy to work in compared to raw HTML+JS. The only downsides are Adobe, the flash runtime, and the intellectual heritage of Java sometimes getting in the way.

Flex is basically, a command-line toolkit to build SWF, while Flash is a GUI-based tool. Flex has the builder based on Eclipse and borrowed lots of conventions from the standard web technologies.

As the browser has been heading towards the capabilities of Flash, Macromedia/Adobe long ago added a layer on top of Flash for application development similar to HTML called Flex/MXML.