The term Flash can include any of the other terms defined below, and I find that saying "Flash" without specifying exactly what you mean can be confusing and misleading. Case in point: I'm not sure what you specifically mean when you say "Flash" in your question. Is it Flash Player? The authoring tool? Or the whole collection of technologies that fall under what Adobe calls the "Flash Platform"? To help clear all this up, let me define the technologies involved in creating Flash and Flex content so that we're all using the same terminology here:
Flash Player is a runtime for rich media content in the browser. There is also Flash Lite to run Flash content in mobile devices (although Adobe has announced that they intend the full Flash Player to run on devices in the near future), and Adobe AIR allows content authors to publish desktop applications with web technologies like SWF and HTML.
Flash Professional (often called the Flash authoring tool or the Flash IDE) has traditionally been the primary application used to create content that runs on Flash Player. It is very designer oriented with timelines, layers, and visual drawing tools. Software developers may find this application disorienting and limited compared to other development tools. When someone says, "I built something with Flash", they often mean the Flash authoring tool, but that's not always the case. For that reason, it's good to always clarify to avoid any confusion.
ActionScript is the primary programming language supported by Adobe to target Flash runtimes. The current version is ActionScript 3 (abbreviated as AS3), which has been supported since Flash Player 9. Content created with older versions of ActionScript can still be run in the latest versions Flash Player, but new features are only supported when using ActionScript 3 to create new content.
Flex is a collection of technologies designed to create rich applications that target the Adobe's Flash runtimes. Though saying "Flex" previously had the same ambiguity as "Flash", the Flex brand name is becoming more and more focused on the Flex framework and SDK, described below.
The Flex SDK consists of compilers, a command-line debugger, and the Flex framework. The included compilers are:
1. MXMLC, an ActionScript and MXML compiler to output the final SWF file for deployment to Flash Player.
2. COMPC, a static library compiler for ActionScript that outputs SWC files.
3. ASDOC, a documentation generator built on the compiler technology.
The Flex framework is a collection of ActionScript classes designed to build Rich Internet Applications. It includes things like user interface controls, web services and other connectivity classes, formatters and validators, drag and drop, modal windowing, and component states. These classes are in the mx.* package. Generally, when developers say "Flex" without any clarifying information, we mean the Flex framework and not the product formerly known as Flex Builder.
MXML is an XML-based markup language used by the Flex compilers to make layout and placing components into containers easier. The tree-like structure of XML make the containment hierarchy easier to visualize. MXML is actually converted to ActionScript during the compilation process.
Flash Catalyst is an intermediary tool to convert graphical designs made with Adobe design tools to skins for Flex applications.
Flash Builder (formerly known as Flex Builder) is a development environment that allows developers to build different project types to create SWF files that are deployed to Flash runtimes. It is built on the Eclipse platform and is more familiar to software engineers. Flex projects use the Flex framework. ActionScript projects do not. Adobe has announced that Flash CS5 will bring support stronger integration with Flash Builder and a new project type that supports FLA files, re-enforcing the appropriateness of the name change.
Flash Builder does not replace Flash Professional. Some people who have traditionally used Flash Professional may now choose to use Flash Builder instead. Often, these are software engineers who appreciate or require the advanced development tools offered by Flash Builder or don't work heavily with assets designed in a visual tool. Some developers may write their code in Flash Builder, while choosing to compile their projects in the Flash authoring tool. Often, these developers are also designers, or they are working with other people who are designers. In this situation, there may be many graphical assets created in the Flash authoring tool, and it could be difficult or simply inappropriate to bring them into another environment.
The Flex framework is specifically designed to build applications. It includes many traditional form controls (buttons, lists, datagrids, etc) and much of the code runs on an advanced component framework written in ActionScript. Not everyone is building the sort of content that Flex is designed to create, and Flex does not replace traditional Flash development practices for everyone. It is a better approach for some developers, but may not be right for others. More design-heavy websites, such as those created for movies, music, energy drinks, advertising campaigns, and things like that probably shouldn't use the Flex framework. Similarly, little widgets you put into the sidebar of your website or on your profile in a social networking website may need to be built with pure ActionScript (without the Flex framework) because they require a smaller file size and they probably don't need a big complex component architecture designed for larger applications. When targeting Flash runtimes, your development environment, frameworks, and workflow should be chosen based on your project's requirements and goals.