I have a feeling that some of the responses didn't address the initial question directly, so I decided to post my own. I understand that the question was about the difference between the mod_php
deployment model and the application server deployment model.
In simple words, PHP executes a given script on every request, and the application has no knowledge of what has happened before (unless it is emulated somehow). Moreover even the source code is being parsed on every request (unless you use a bytecode cache like APC). This process can be slow, especially if you have a framework with complex initialization.
In contrast to this, the application server has to be started once, and then it waits for a request to be processed. The application server should clean up resources after every requests (allocated memory, open descriptors, etc.), it can also pool certain resources (like database connections) that can be reused between requests for extra performance.
This later model (application server) is more efficient in most cases, but on the other hand more difficult to setup and maintain. It is also more demanding, as you have to pay more attention to the resources you utilize, in order to avoid resource leaks.