There are many similarities, many places where their functionality overlaps. I’ve always been tempted to think these libraries are interchangeable, but I’ve come to realize that’s not the case. They have fundamentally different goals.
Which one is better? I can’t really say. They both have benefits. I like MooTools’ dynamic element creation and manipulation better, but jQuery has a much larger and more active community. jQuery is easier to learn and the documentation is much better, but MooTools provides an inheritance mechanism which is very nice if you’re used to OOP.