Greatest strength of jQuery is DOM traversal and manipulation. These have been made so easy with jQuery that one often tends to slack off when writing markup, thinking how easy it is to manipulate it using jQuery. In addition, jQuery also has excellent list of plugins and jQuery UI which is great for extending the capabilities of jQuery and designing awesome user interfaces.
Moo Tools has a plugin directory of its own, it may not be as rich as jQuery’s plugin directory but it is certainly one you can turn to when in need. The documentation is good and there are books that focus on Moo Tools so you can get started and be up to speed easily.
Dojo Toolkit also offers rich UI widgets that you can use to enhance the user interface of your application. Dojo supports all the popular browsers and the documentation is good along with a number of books that you can buy to learn the ins and outs of Dojo Toolkit.
If it is the user interface you are focusing on, when developing your web app, Ext JS is perhaps one of the best solutions out there. Ext JS provides a comprehensive set of widgets which give your web app a desktop application like look and feel. The UI widgets are rich in functionality as well, capable of fetching data via a variety of methods and displaying them exactly like you want.
The library features number of UI widgets, utilities to take care of animation, drag and drop, AJAX calls and the likes. Contrary to other popular libraries, YUI actually lets you choose just the features you want to use, which keeps your application lightweight. You can find a lot of examples and tutorials, along with excellent documentation to help you along.