I know both languages are from the same ECMA-262 standard. It seems that the two are becoming very similar with JavaScript adding event listeners for core Object instances t
I've been programming in both ActionScript and Javascript, and from a less-technical standpoint, I see two main differences.
1) JavaScript is more powerful. You are allowed to do much more with the language because it does not have a "compiler" or types. There are some great frameworks out there like ExtJS and jQuery that try and simplify things for you, but even with them, you are really allowed to do an amazing amount of damage if you want to.
2) ActionScript is much more confining and hence, much easier to maintain. Adobe did a lot of work to keep you out of the difficult parts of ECMAScript. ECMAScript Objects, prototypal inheritance, and closures are three concepts that you really don't need to understand to program in ActionScript. You just need to understand how to use Adobe's "Class" object.
For simple uses, I prefer JavaScript. However, once the project gets large, it depends who you are coding for. If I had a team of 5 developers programming at a scrappy start-up, I'd choose JavaScript in a heartbeat. However, in the girth of a large corporation, or academia, you might be safer relying on Adobe's platform.
Hope that helps.