My bank doesn't recognize my computer on FF but does on IE.
When I attempt to log in to my banking account the bank doesn't recognize my computer. I have told the bank system that this is a private computer on earlier logins yet it asks for additional authentication each time. This does not occur when I use Internet Explorer, only with FireFox.
Opaite Mbohovái (2)
I don't know what kind of security your bank requires, But if your FF 34 was updated it might have SSLv3 turned of.
See Mozilla's security blog The poodle attack and the end of SSLv3
Basically, the web site places a cookie so that it knows that computer. BUT, by using a different browser, you are now using different cookies. If there is something on the page to remember that computer, try using it. But I don't know if it works for two browsers.