This catches me for a moment every time I start working from a new location on my disk: If a local SWF plays in the standalone Flash player, but does not run in Web browsers, it may be a security issue. The Flash browser plugin by default does not trust local SWF files and won’t execute them. Unfortunately, somewhat ridiculously, it does not issue any sort of alert or indication that it is blocking the file for this reason, so you may find yourself just sitting there wondering what’s going on.
Right click on the Flash stage in the browser. In Global Settings -> Advanced -> (Developer Tools) Trusted Location Settings add either your specific SWF file or a parent directory. I usually add an umbrella project directory, e.g., /home/joe/rocketshipgames/haxe/. The SWF should now play when you reload the page.
Alternatively, you could set up a local webserver and access the SWF through that rather than a file URL; Flash by default trusts localhost.