Ext JS is a JavaScript application framework to build interactive cross-platform web applications using Ajax, DHTML, and DOM scripting. JQuery, on the other hand, is designed to simplify HTML DOM, Ajax, and CSS animation that acts as a Javascript library to perform such a task. ExtJS cannot accommodate the traffic on sites on the Internet unlike JQuery that can handle websites, thus, the choice and most used application to run a website.
ExtJS is often used in corporate websites where only the employees can use it. Javascript is used in ExtJS while in JQuery a user can write in HTML code. Corporates choose Ext JS over JQuery because of its rigidity and offer a good set of built-in widgets than JQuery.
These two things are usually used to develop new applications or new software. There are a lot of people who love using JQuery because this is known to be fast, classy, and also secure. ExtJS is also known to be used more in creating plug-ins or widgets. This is also known to be cheaper than JQuery.
This is known to be easy to use, but some people feel that they do not get a return of the money that they spend on it immediately as compared to using JQuery, which is better known. A lot of people are also familiar with the JQuery as compared to the ExtJS, but it would still be up to you which one you would instead use.