what's the difference between using a js library like jquery and using an ajax framework what's the most active ajax framework out there( preferably for PHP)
or a better question would be...what's the difference between an ajax request fired with Jquery and an ajax request fired using any "AJAX frameworks"
what's the difference between using a js library like jquery and using an ajax framework like Akelos?
I believe Akelos is a PHP framework and NOT a Javascript framework.
As PHP frameworks typically sit on the web server and generate web pages that are then rendered by the client, a PHP "AJAX" framework is likely going to generate JavaScript code that is then executed by the web browser. The JS code that gets generated is serving the same role as JS you would write by hand to do the same thing, such as the jQuery.ajax() call you mention.
So, typically, the difference is between code written for you, and code you write by hand.
I think you are confused about what is Ajax and how it is related to PHP. Basically "Ajax" just means that some javascript code is making calls behind the curtains to some backend server to make an action or update something on the user interface.
That backend server language could be PHP, ASP, Ruby or anything really. Thats why the most popular "Ajax Frameworks" (Prototype, JQuery) are purely Javascript, and don't depend on you using PHP or anything.
On the other hand there's PHP Frameworks like CakePHP, CodeIgniter, Zend Framework (or the one you mentioned, Akelos) who just help you doing some tedious tasks in PHP, like templating, database access, etc. These frameworks don't depend on any "Ajax Framework" you can use any of them.
So just look at this separately, there's "PHP Frameworks" and there's "Javascript Frameworks" aka (badly) "Ajax Frameworks".
Of course there's always the exception to the rule, there's some project who make PHP calls to output Javascript code, like Xajax. But i would recommend you to avoid them or you will end up more confused.