I had this happen to me as well.
The twitter and outlook issue are the same thing, lets do some troublshooting.
1. Power down your phone, and power it back on. Turn off Wifi and ensure you have a data connection. Go to your mail and sync the account on your phone. (this might fix it), if it doesn't work go to airplane mode and turn on wifi and try the sync this way to force it over the wifi network.
2. Go to a PC, and log into your outlook.com account. Does it prompt you to change your password. Your account may have been hijacked and one of your contacts may have said your account was stolen. If it doesn't prompt you I would change your password, then go to your mail on your phone click sync, it should prompt you for your new password.
One of these should fix your issue, what error message are you getting?