jQuery is like the ASP.NET AJAX Client Framework (MicrosoftAjax.js), with selectors, DOM selections/ manipulations, plug-ins, and better animation support. jQuery is more powerful than MS AJAX on the client side due to its light weight nature . This is the reason Microsoft integrated jQuery with Visual Studio. JQuery is integrated for post VS versions 2008, no explicit download of jQuery file is required for the versions above VS2010.
Ajax is a Technology for Asynchronous Data Transferring. AJAX is a technique to do an XMLHttpRequest from a web page to the server and send or receive data to be used on the web page.
jQuery can be used
If you love and comfortable with JavaScript
Most interaction is client-side only
If a custom solution is required
For stunning look and feel of client side UI
If animations ,DOM Selection are required
Ajax can be used
If you are using ASP.NET & VS
When server side Integration is required.
If you need json and WCF Support
Read more: Beyond Realtional