Are you talking about the Flash player (end user perspective) or Flash/Flex development tools (developer perspective)? From the end user perspective, there is no difference between Linux and Windows at all. There was one, many years ago, when the latest version of Flash Player for Linux was always significantly behind the Windows release, but this is a thing of the past. For several years now, the most up-to-date version number for Linux is always the same as the Windows version, new versions are released for both platforms at the same time, and I suppose they are more or less identical. Furthermore, graphics drivers (and other performance-critical software components) for Linux have improved a lot, so you do not only get the same features, but similar performance as well.

From the developer perspective, the situation isn't that good. AFAIK, there is only one really good Flash/Flex IDE for Linux on the market, that's FDT by Powerflasher Solutions, a commercial product that you have to pay for (although it is probably worth the money). For the Windows platform, you get FlashDevelop - which is really great - for free, and then there are Adobe's "official" tools, of course. I'm both Linux enthusiast and part-time Flex developer, but I hesitated to buy FDT so far and use FlashDevelop on Windows instead. However, this might change in the future.

You can do flash/flex development on linux using the command-line compiler (it's in the free sdk, check Adobe's site).
This worked well for me back when I used to do flash/flex development. Of course, I'm the type who uses vim for everything... if you prefer IDEs more like Visual Studio, Eclipse, etc. then the vanilla compiler might not be useful for you.

I know about the free Flex SDK, of course. I've used it under Linux before, and I also use it under Windows with FlashDevelop. The compiler is not an issue, but editors are. I appreciate comfort features like code completion or "go to declaration of some function" a lot, they give incredible productivity boosts. And AFAIK, FDT is currently the only Flex editor that has such features and runs under Linux without problems.

Vim supports autocompletion and jumping to definitions with the right scripts/plugins installed. Actionscript isn't the best-supported language in this respect, but there's some stuff out there, such as: http://www.vim.org/scripts/script.php?script_id=2909. And if you're not in a hurry, you can make your own .

Of course, vim itself takes a while to get used to if you're new to it. However, I consider it worth the investment: even without autocompletion and other language-specific features, I find editing in vim to be quicker than most IDEs (there are non-language-specific productivity boosters that can apply to all general text editing).

It's not everybody's cup of tea, but it's an option.