Both Flex and AJAX are client-side technologies and both can be used to create RIAs or to “augment” existent web applications. Like in the case of Flex, typically an AJAX application makes a request to server after the initial response was rendered, and injects the response into the existing page once the servers sent the answer to that call.
While Flex is an application framework to build apps for Flash Player or Adobe AIR, AJAX is a term that refers to a collection of technologies and techniques that include JavaScript, HTML, and CSS. Typically when you work with AJAX you use a framework (Ext JS, Dojo, jQuery, Prototype and so on). The reason you are doing this is mainly to cope with the differences between browsers implementations for JavaScript/HTML/CSS.
Having said that, in the end it is really up to you what technology to choose. If you prefer to work with a dynamic, weak-type, prototype-based language, you will probably favor AJAX. If you have experience with JavaScript and you don’t want to learn a new language, again AJAX is the answer for you. If you prefer strong-type, OOP languages, modern IDEs features you’d probably prefer Flex. Arguably Flex (and Flash Player) offers more flexibility and features when compared to what is possible with AJAX (even with the new additions of HTML 5). But again, there is no such thing as one is better than the other. It really depends on the nature of the project you are working on, your experience, etc.
In fact, there are people using both technologies to do things that would be impossible to do with only one of them.