The AIR app won't work in a browser. For the web version, you would use Flex and for the desktop version you would use AIR. Both applications can be written in Actionscript 3 however, unlike Flex apps, AIR apps can also be written in HTML/JS. If you write your apps in Actionscript 3, your Flex and AIR apps could use almost identical code. Obviously, there are a few classes which can only be used in AIR as these related to desktop functionality. If you look at the Actionscript 3 reference (http://help.adobe.com/en_US/FlashPlatform/reference/actionscript/3/), you can see which of the classes are AIR only and which can be used in both Flex and AIR.
AIR files are exported as .air files while Flex file are exported as regular Flash .swf files.
As far as, "C++ (which is Air) and i could use PHP (which is Flex)", that is sort of right. AIR is a replacement for C++ in that you can write desktop apps with both. However, because of the AIR runtime, you do not have to write different versions for different operating systems. the one AIR app will work on Windows, Mac, Linux and now Android phones. With C++, you'd have to modify your codebase for all of these environments. Flex is not like PHP however, it's client side so is more like a more-powerful Javascript. You can have a Flex app talking to a PHP backend (using AMFPHP or Zend_AMF to communicate) which handles all the server-side stuff like retrieving data from a database and sending it back to the Flex app running in the browser on the client. Flex is sort of like a collection of components (it's officially an SDK) to make writing web apps easier. So, for example, there's drop-down menus, date-pickers, datagrids, trees, video displays, etc, plus everything else that's available in Flash. IMO, the best thing about Flex development over HTML/JS development is the layout model. You can make some very complex fluid layouts very easily in Flex which adapt to different screen sizes. You also don't have to do much in the way of cross-browser testing - it looks the same in all browsers on all OSes as long as the user has the approriate Flash Player. It's also very fast when handling lots of data and makes it easy to mix many different multimedia components, video, audio, etc, in the same app. Some good apps written in Flex are Aviary, Photoshop.com and Buzzword. Some good AIR apps are the eBay desktop client, Feedalizr, Seesmic Desktop, etc.