My company develops several types of applications. A lot of our business comes from doing multimedia-type apps, typically done in Flash. However, now that side of the hous
You seriously shouldn't use ANY of these ActiveX2.0 technologies. Neither Silverlight nor Flex...
First of all, both of them are nothing more then "distributed winforms frameworks with support for being ran in the browser", second of all they don't port well to other devices (specially true for Silverlight), thirdly they don't work good with other portions of your page. They don't work well for disabled people, etc, etc, etc. List goes on into infinity...
Adobe and Microsoft both tries to hide this fact really hard, but at the end of the day both Silverlight and Flex is nothing but ActiveX in a new wrapping...
Sure they run in sandboxes, are managed languages and all that. But it's still a big piece of BLOB being downloaded to run locally in your browser, AKA ActiveX...
You will never get an fair vote to this question on SO as it has so many Microsoft devs.
Also, people will probably down vote this answer, which says it all really.
I say let your developers try both platforms, and see which they prefer.
To answer the comments below, I just noticed that while there are lots of answers recommending Flash / Flex, the ones for Silverlight have many more up votes. It's not a matter of lying, it's just favouring what you're familiar with, not necessarily the best platform.
My team used to write rich web features in Flex, and now writes them in Silverlight.
Our reasons for this switch:
The rest of our project is .Net and C#, you may find all these issues less significant in a Java shop.
Another advantage of Flex development is that you can switch to developing desktop applications (Adobe AIR) with the same source code (and same IDE) and distribute them from web. You can check out this
for the future of Flash platform.
Update Q3/2011: Flash 11 supports low-level 3D acceleration, and there are already many frameworks and major engines (Unreal Engine 3, Unity) supporting it. The selling point for the future, however, is that AIR application will work on Windows, Mac, Android, Playbook, and iOS platforms (Linux support has been dropped). With an absolute minimum of hassle between porting between those (at least when you have Adobe CS5.5+).
Update Q2/2015: Silverlight is officially dead. Adobe AIR is alive, but not thriving - it might be useful based on your skills and tool chain. Both Microsoft and Adobe admit that HTML5 is the way to go (whether with AIR or Apache Cordova or Visual Studio).
Update Q3/2017: Haha wow, who even uses Flash anymore.