Internet Explorer/FireFox Flash Player difference?

There frequently tend to be posts “My SWF works in FireFox but not Internet Explorer! Help!!”

I’m wondering, is there REALLY ever any truth to these and not just silly mistakes?
**Is there any real difference between the Flash Player in Internet Explorer and FireFox? **

Aren’t they the same, and all the browser is doing is relaying those commands to the library/executable already on the computer? The reason I’m guessing this is because when you install Flash Player, it works for all installed browsers, and you don’t have to install it to specific browsers, right? Or?