Fifth:- Taraweeh prayer has been called قيام رمضان in the Hadeeth Shareef. Meaning Taraweeh has only been specified for Ramadan. And Tahajjud has been called (صلاة لليل) or night prayer. Meaning, Tahajjud prayer has been assigned for the whole year and not for Ramadan specifically. Therefore, this proves once again that Taraweeh and Tahajjud prayers are different.
The order to perform Tahajjud prayer has been given from Makkah Shareef, and the order for Taraweeh has been given from Madina Shareef.