The big difference between the two is the development environment, Flash is meant for designers, they have a timeline, drawing tools and palettes for choosing color and drawing tools behaviors. While Flex consists of a more programming environment like a general programming language and a component list (controls). They both compile into SWF files and both can access database data or make other forms of communication.

Flex also contains MXML (Magic eXtensible Markup Language) for defining the GUI layout which is not available within Flash.

<?xml version="1.0" encoding="utf-8"?>
<mx:Application xmlns:mx="http://www.adobe.com/2006/mxml">
    <mx:Array id="sampleArray">
        <mx:String>Sample Label 1</mx:String>
        <mx:String>Sample Label 2</mx:String>
    </mx:Array>
    <mx:Panel title="Example Panel">
        <mx:ComboBox dataProvider="{sampleArray}"></mx:ComboBox>
    </mx:Panel>
</mx:Application>

It is however possible two use both at the same time. Designers can create animation and visual elements and developers can integrate this
within their Flex application.

Both environments can run in a browser, however Flex also offers the ability to be started directly from the desktop, for this you need the 
Adobe AIR application.

there are more differences and there is more information which can be found in the following video:

Flex in a Week: Comparing Flash, Flex, Flash Player and AIR