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
Conclusion
jQuery and Ajax often used in conjunction with each other for better results.
jQuery is  used to modify data on the webpage dynamically and it  can use AJAX to retrieve data that needs without changing the current state of the displayed page through partial post backs