It seems that there is a recent problem with SWFObject in Firefox: SWFObject does not load Flash, and it just keeps displaying the alternative content for browsers without Flash enabled. Upgrading Flash player does not solve this Firefox specific problem.
The ironic part is that it seems that the source code of SWFObject has not been changed recently. So this bug is related with Firefox, and most probably, with a corrupted automatic Flash Player update that conflicts with SWFObject just in Firefox. A normal reinstallation of the latest Flash Player does not seem to solve the problem.
One way to solve specific SWF Object problems in Firefox is disabling the automatic hiding of the alternate content till the Flash content is fully loaded. This bug has been reported for Firefox 3.5 and is solved just by adding swfobject.switchOffAutoHideShow() in the SWFObject code.
If all the previous fixes fail, the only way to solve this SWFObject problem in Firefox thus far would be reverting to a very old SWFObject version. Both SWFObject 2.1 and the recent SWFObject 2.2 versions seem to be experiencing this problem in Firefox, but versions from when SWFObject was still called FlashObject seem to work fine.
If this problem is not solved quickly it would be better to remove SWFObject from the webpage, or probably several Firefox users won't be accessing the Flash content. The latest posts on this bug tracking thread seem to be closely related to this bug. Let's keep an eye on it till the fix is found.