Web UI development tools
I agree with the other popular answers, but our needs were for a JS library on which we would develop our own code. Other libraries may be better suited for rapid UI development or use on projects that will involve different teams. For us, we have a single web-based product that has extensive proprietary JS development (about 41k lines of JS code). All the popular libraries, MooTools included, help deal with the DOM and browser issues. More important for us is working with objects, functions, and scope.
1. MooTools takes a prototype approach of extending native objects. I think this results in nicer code in the end. You don't need to wrap everything in the big code object like you do with jQuery. The jQuery way is better if you are working with multiple libraries and for us that is not a concern. This is the most fundamental difference between jQuery and MooTools.
2. MooTools has excellent code for extending objects, working with arrays, dealing with variable scope, events, and functions. In other words, the fundamental pieces on which we build our own code.
3. I like the design of the MooTools Element object and its functions, especially Element creation and the function pair store and retrieve.
4. The MooTools team have done a good job of improving the library while also making tough decisions about the structure and function names of the library. The last few upgrades have been painless thanks to their compatibility tools, but the 1.1 upgrade was rough.
5. We use almost no 3rd party code, so having a database of widgets and plug-ins is not a concern. However, MooTools is popular enough that it is one of the two libraries supported by the excellent Highcharts library (the other is jQuery). That said, MooTools does have a code gallery:
If you are building an application where you will be writing a lot fo JS I think MooTools is the best decision. We went that way in 2007 and I have no regrets.