The largest difference I see between Prototype and Ajax is Prototype feels more like an Object Oriented approach where JQuery feels more like an expression language. Consider this...
Prototype ----------- new Ajax.Updater(...)
JQuery ----------- $.load(...)
Both do the exact same thing, more or less. They take the response from an AJAX request and shove it on the page.
I think both are strong and well-written libraries, and choosing either depends upon which approach you prefer. Personally, I have adopted jQuery for use in all my personal projects as I like its lighter, non-invasive style (and there are tons and tons of jQuery plugins to do just about anything one might need). In my day job, we use both. [ January 16, 2008: Message edited by: Bear Bibeault ]
jQuery was inspired by a number of things, including Dean Edwards' Base and the original Behaviours library that was written on top of Prototype. John is also friendly
Prototype's primary mission is to make JS more like Ruby; jQuery's primary mission is to make it easier to do DOM Scripting. See other threads from this week for more details.
There's a lot of jQuery's approach that has been adopted by Prototype. Most notably their revamped event system looks a lot like jQuery's event system. Then again, pretty much everyone has come on board with things like full CSS3 selectors and simplified events.