When looking at Ext JS, I don't get the feeling that it's meant for the same things as jQuery and Mootools.
Whilst jQuery and Mootools help with the general workings of a site, Ext JS seems to be focussed on tables and storing data, plus manipulating it.
So, is this observation correct, or is Ext Js fit for the same work as jQuery and Mootools?
Ext JS is a full-fledged widget library while jQuery (not jQuery UI) and Mootools are JavaScript frameworks that help with DOM manipulation etc. Your observation is correct.
ExtJs and JQuery are kind of apples and oranges. You can compare Ext Core to JQuery, and ExtJs to JQuery UI.
Ext Core, and ExtJs, are Open Source, with Core being under an MIT license and ExtJs falling under the GPL 3. Both are ultimately managed by Ext, LLC, including all changes to the libraries. Ext has a dedicated development team and support staff, and those utilizing the commercial license with a support contract have full support for help and issue resolution. Aside from that, the Ext control classes (and all other classes) are very comprehensive and consistent.
All of the libraries you mentioned have there pluses and minuses, but what I've already mentioned helps make ExtJs a compelling choice for me.
We used ExtJS to build, very rapidly, a sophisticated call centre application with almost no concern for styling, as this all comes "out of the box". However, ExtJS uses a more traditional UI, so if you want more fine control over UI or want something flashy or very unconventional, then Ext is not the right choice. Mootools and JQuery are better for original design but require much more effort in getting the styling right.
Note that ExtJS can be used with other libraries and we do use it with JQuery, simply because we have different apps, built in different libraries, that must sit together.