from Wikipedia: In 2013, Hotmail was replaced with Outlook.com, which features Microsoft's Metro design language, and closely mimics the interface of Microsoft Outlook.

Yes I know about the change to Outlook. My problem is that I have a Hotmail email and an Outlook email. Firefox doesn't allow me to get my Hotmail email.

This could be a problem with a cookie or a saved login:

• As long as you have a cookie keeping you logged in to account #1, Outlook won't show the login screen and let you sign into account #2 until you sign out under account #1.

• If you had Firefox save your login for account #1, then you might not be able to access the login screen and sign in with account #2. To clear a saved login, you can use the Security panel of the Options dialog here:
  • "3-bar" menu button (or Tools menu) > Security > Saved Passwords

If you need to access two accounts on one website simultaneously, you might be able to use a regular browsing window for one and a private browsing window for the other. In a private window, Firefox won't use your cookies from the regular window. To open a private window, you can use the menu or press Ctrl+Shift+p.