I have a Mac and a PC and have the same problem on both! I end up forcequitting on the Mac and using Task Manager on the PC. This has been going on for 2 weeks. Why does this happen, I’m almost ready to give up on FF.
Firefox doesn't close or when you open it again the Windows appear again. To make Firefox open a fresh window, go Tools --> Options --> General --> and select from the drop down box "Show Blank Page"
is it a raw Firefox installation on both computers or do you use the same addons on both? Did you try to disable all addons? There may be a conflict.
If disabling all addons fixes the problem you can turn them back on one after the other and you will eventually find out which one is causing the issue.
Also try the tips in this guide: