Edits: Replace "Flex" with "Flash Builder", replace "Flash" with "Flash CS", and add the following comment: This post is really about Flash CS5 and Flash Builder (formerly Flex Builder). See the other answer to this question for a good discussion of the Flex framework itself.
You might think of Flash CS and Flash Builder as two IDE's for working with ActionScript (AS). If you want to make drawings and animations, Flash CS gives you much better GUI tools for that. If you want to make applications that are more data-oriented, with things like forms, datagrids (like tables), and the like, Flash Builder has better tools for that. Ultimately, each is providing a set of UI tools which result in a (more or less hidden) set of AS classes that are then compiled into a SWF (movie playable by the flash player). And once you get down to the level of directly writing AS classes, I'm pretty sure you can do anything in either that you can do in the other. So it's kind of a question of which gui is better suited to your needs. Also, last I looked anyway, Flash Builder is built on top of Eclipse whereas Flash CS is its own standalone application. If you like Eclipse, Flash Builder has an edge; obviously, if you hate eclipse, the opposite is likely true. One nice thing is that Eclipse is still eclipse so you can treat AS as just one of the languages it supports: i.e. you could also use it for Java, ColdFusion, PHP, etc.
Unless this changed in the latest version, I'm pretty sure that the one thing Flash Builder can do that Flash CS can't is build an AIR app.
Books. A good AS book should teach you about the extensive class library that's built into AS, and how to extend it with your own packages and classes. It could vary from pretty much a printed version of the language reference that Adobe has online, to a book about programming in AS, to a book about design patterns in AS. It might teach you anything from just about nothing to a fair amount about how to use the Flash CS or Flash Builder IDE. On the other hand, a good Flash CS or Flash Builder book should teach you a lot about how to use the IDE, including its various user interfaces, and may cover anything from a little to a great deal of AS.
Finally, unless you're committed to a book, you might just start with a free 30 day demo, and the extensive built in help. Beyond that, there's lots of stuff available on Adobe's website. And then there's all those websites about coding and flash, etc.