I told Glitch that I forgot my password, and it let me choose a new one. After you set a new one, you'll have to log in again with your new password on your working computer, but logins from other computers should work afterwards.
That's a temporary workaround until the devs do something about it at least.
You don't actually login with your Facebook authentication, you should just be using the same email address your invite was sent to. Can you try logging in from the other computer with the email/password you set up for the account on the first computer? That's the standard Way Of Doing Things, so it should work, and if it doesn't we'll need to figure out what's going on.
After you click the Facebook link, you come to a page with 4 fields: Your name, Your email, Password, Verify Password. The name and email fields are populated when you get there, and then you enter a password.
alts: It's possible this form's changed since you first registered, if you registered a while ago, but if you don't remember your password then you can do what Dryea did and get a password recovery. Try either the email you got your invite code at, or the email you use for Facebook, depending on how the form got filled out.