Sounds like a pretty simple case of some corrupted cookie files. Also possible you installed some kind of security related add-on to the browser recently, and that's part of the problem. Good place to start, is comparing every difference between the browser on the computer that works and the same browser on the computer that doesn't work. You can also download a second, or third, web browser for the computer not working. If that one lets you log in properly, then your issue is with the browser config. If not then you probably installed some new AV program or something that is screwing up your access.