I've had this problem on and off of FF continuing to run. Usually I can start it OK when I first boot, but if I close it and try to open it again too soon, it doesn't. Task Manager would show 2 instances of it running. I kill the first one, it closes, and then either both will close or FF opens. After that, no more problems. If I leave it alone, minutes later, it opens anyway. Something is hanging the first instance open for me. I've learned to live with it.
It NEVER effects IE or the use of the Internet/network for me though?
What version of FF are you running? I never get that dialog box either?
Are you running x86 or x64 version of Vista?
Have you tried running FF in SAFE MODE, that is without all the add-ons? It could very well be an add-on or even a PROFILE that is causing the problem.
Instructions to run FF with a new Profile as a test is HERE, give it a try with a new one and see what happens...
To start FF in Safe Mode, not running in Windows Safe Mode, check HERE.
Let us know what you discover?