Do you have automatic user/pass authentication enabled on both browsers?
First, you should clear the browser cache (cookies, temp files etc), and clear all saved user/pass credentials.
This may differ according to browsers, on IE it is on
Tools -> Internet Options
-> "General" tab to remove cookies and temp files
-> "Content" tab to clear Autocomplete data (stored usernames & passwords)
Note that clearing saved user/pass data will erase ALL saved authentication credentials, so you'll have to type them again on every site you were eventually logging in with autocompleted user & pass.
As a start, it's always worth a try...