My opinion on this one's pretty simple: unless the app needs to be accessible publicly, unless it needs to be search-engine optimized and findable, and/or there's an otherwise compelling case for its having to remain strictly text-based, then the chips are stacked in favor of rich-client runtimes like Flash or Silverlight right out of the gate.
A big reason, if not the biggest, is that they eliminate the complexities of developing for multiple browsers and platforms. Again: they eliminate the runtime-environment variable. No more debugging old versions of Netscape and IE, no more object detection and consequent branching, no more wacky CSS hacks -- one codebase, and you're done. Offloading the runtime environment to Adobe or Microsoft will save you time, money and headaches, all else equal. (Sure, there's YUI, JQuery, etc., but they don't eliminate that variable -- they just abstract it. And they don't abstract all of it, either -- only some of it; ultimately, it's still up to you to test, debug, retest, debug, repeat.)
Of course, your situation's a bit more complicated by the existing-codebase problem, and it's difficult to say definitively which way you should go, because only you've got the code, and we're just geeks with opinions. But assuming, just by your asking the question, that a refactoring of your existing codebase would involve a significant-enough undertaking as to warrant even considering an alternative (and probably comparatively foreign) technology in the first place, which it sounds like it would, then my response is that your curiosity is well-placed, and that you should give them both a serious look before making a decision.
For my part, I'm a longtime server-side guy, ASP.NET/C# for the past several years, and I've written many a text-based line-of-business app in my time, the last few with strong emphasis on delivering rich soverign UIs with JavaScript. I've also spent the last couple of years almost exclusively with Flex. I've got experience in both worlds. And I can tell you without hesitation that right now, it's everyone else's job to beat Flex: it's just an amazingly versatile and productive product, and for line-of-business apps, it remains leaps and bounds ahead of Silverlight. I just can't recommend it highly enough; the data-binding and event-handling features alone are incredible time-savers, to say nothing of the complete freedom you'll have over layout, animation, etc. The list goes on.
So, my advice: take a long, careful look at Flex. In the end, you might find a ground-up rewrite is just too massive an undertaking to justify, and that's fine -- only you can make that determination. (And to be fair, you've just as much ability to make a mess of a Flex project as you do with a JavaScript project -- I know. I've done it.) But all told, Flex is probably the least-limiting, most flexible, most feature-rich and productive option out there right now, so it's certainly worth considering. Good luck!