Is there a way to have Flash work in both Firefox and IE at the same time?

I’m a longtime (and unapologetic) IE user, and when I use Firefox, Adobe Flash Player just doesn’t work. Lately, I’ve been trying to use Firefox more frequently, so I uninstalled Flash, then installed it from Firefox, and now it works great in Firefox, but won’t work in IE. I did this because it was recommended on various forum sites.

But, does anyone know why this is? And, is there a way to get Flash to play nice with both IE and Firefox, and not choose sides?