To confirm, you are using an Outlook.com account on your phone and in outlook desktop, correct? (The primary address needs to be outlook.com, Hotmail, live or msn domain.)
Is the account configured in Outlook desktop as a Microsoft Exchange account? Go to File > Account Settings > Account Settings to open the Account Settings dialog. The account type is to the right of the account name.
If it is an exchange account, do new tasks in outlook sync up to outlook.com and into to-do app?