Short answer
In the Hanafi school they are two different prayers. Taraweeh is a sunnah mu'akkadah of Ramadan nights, prayed after Isha, usually in congregation, 20 rakat. Tahajjud is the night prayer of the whole year, best prayed after some sleep in the last part of the night. In Ramadan you can pray both.
Taraweeh is a separate sunnah mu'akkadah particular to Ramadan, prayed after Isha and before or after witr. Tahajjud is a sunnah (many texts say mustahabb) prayed after sleeping, all year. Praying taraweeh does not replace tahajjud.
The four schools agree on this.
Explanation
Both are prayers of the night, so the question is natural. The Hanafi scholars, and the four schools in general, treat them as two distinct prayers.
- Taraweeh: only in Ramadan. Its time is after Isha until dawn. It is sunnah mu'akkadah, prayed in congregation in the masjid as a sunnah of the community, 20 rakat in sets of two, with a rest after every four. It has its own name and its own practice from the time of Umar.
- Tahajjud: all year round. It is best after waking from sleep, in the last third of the night, when Allah descends to the lowest heaven and asks who is calling on Him. It is usually prayed alone, two rakat at a time, as many as you can.
Can you pray both? Yes. Many people pray taraweeh after Isha, sleep, and wake before suhoor for a few rakat of tahajjud. If you have prayed witr after taraweeh, you may still pray tahajjud later, but do not pray witr a second time. If you are confident of waking, you may delay witr until after tahajjud.
The other view. Some scholars today hold that taraweeh and tahajjud are the same night prayer, citing Aisha's report that the Prophet ﷺ did not pray more than eleven rakat at night in Ramadan or outside it. The Hanafi scholars understand that report as describing his tahajjud through the year, while taraweeh is the Ramadan prayer that the Companions prayed in congregation. Keep this disagreement respectful.
For more, see is taraweeh sunnah, is tahajjud sunnah and how to pray tahajjud.
The evidence
Quran, Surah Al-Israa 17:79
وَمِنَ ٱلَّيْلِ فَتَهَجَّدْ بِهِۦ نَافِلَةًۭ لَّكَ عَسَىٰٓ أَن يَبْعَثَكَ رَبُّكَ مَقَامًۭا مَّحْمُودًۭا ﴿٧٩﴾
And from [part of] the night, pray with it as additional [worship] for you; it is expected that your Lord will resurrect you to a praised station.
Tahajjud is the voluntary prayer of the night, praised in the Quran.
Sahih al-Bukhari 1147
Sahih... ما كان رسول الله صلى الله عليه وسلم يزيد في رمضان ولا في غيره على إحدى عشرة ركعة ...
"The Messenger of Allah did not pray more than eleven rakat, in Ramadan or at any other time."
This is the report the other view relies on; the Hanafi scholars read it as his tahajjud.
Sahih al-Bukhari 1145
Sahih... ينزل ربنا تبارك وتعالى كل ليلة إلى السماء الدنيا ...
"Our Lord descends every night to the lowest heaven when the last third of the night remains."
The last part of the night is the special time of tahajjud.
Where the ruling is recorded
- Radd al-Muhtar (Ibn Abidin), Kitab al-Salah, chapter on witr and nawafil. Taraweeh and tahajjud are distinct; tahajjud is prayed after sleep.
- al-Hidaya, Kitab al-Salah, chapter on the night prayer of Ramadan. Taraweeh is a sunnah of Ramadan prayed after Isha in congregation.
Checked by our team. For a more accurate answer to your own situation, consult a scholar. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.