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.
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.
Comparing ExtJs and JQuery is like comparing 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.
I think, ExtJS and jQuery is completely different. Ext gives yor power of UI Components. Core jQuery -not with plugins- gives you speed and very very useful for DOM Manipulating and quick animations.
If my application needs a Grid, Window, TabPanel or TreePanel, the framework should be ExtJS. In this case, I do not deal with jQuery UI or jQuery UI Plugins.
If my application does not need a Grid, Window, TabPanel or TreePanel, the framework should be jQuery because script size really important and jQuery is easy to use for such an application.
If my application needs a datepicker, colorpicker, basic tabs or basic accordion panel, the framework should be jQuery + jQuery UI. Because script size and usage.
ExtJS would be a great approach for admin panels that uses widgets highly. Have a look at what ExtJS can do ExtJS Demo Page