The meaning of Tahajjud:-
The word Tahajjud means-prayer at night, and the person who wakes up at night to pray is called a Mutahajjid. Imam Azhari says, in Arabic a person who sleeps at night is called a Hajid, and the one who gets up to pray is called a Mutahajjid. (Misbahul Logaat Page-977, also found in Lisanul Arb, Kamus Al Muhit and Al Munjid)
‘Tahajjud’ means – Staying up at night, the prayer which is performed during midnight. (Firozul Lugaat, Page-393)
Therefore, it is clearly proven by the definition of “Taraweeh” and “Tahajjud” that Taraweeh prayer and Tahajjud prayer aren’t the same, rather they are very different from each other since Taraweeh’s time is after Isha, meaning it can be performed at any time from midnight till dawn.
Tahajjud prayer on the other hand won’t be accepted if it’s performed before midnight. Rather it will be counted as Nafl prayer. Meaning that the time for Tahajjud prayer is from midnight till dawn. It is best, however, to perform Tahajjud prayer after waking up at midnight.
Mainly, Ha’drwat Umr Ibnul Khawab Rwadiallahu Anhu (can’t understand)
والتسى تنامون عنها افضل منالتى تقو مون يريد اخر لليل وكان الناس يقومون اوله.(بخارى شريف
Meaning:- “It is better to pray during the time you sleep. (Abdur Rahman Rwahmatullahi Alaihi says). And by this he meant the last part of night since (durin Ha’drwat Unr Ibnul Khwab Rwadiallahu Anhu’s time) people used to pray Taraweeh during the first night.” (Bukhaari Shareef)
Meaning, it was better to perform Taraweeh during the last hour of the night, but for the welfare of the Ummate Habeebi, he brought the Taraweeh prayer during the first hour.
Therefore, praying Taraweeh regularly after Isha prayer started from Ha’drwat Unr Ibnul Khwab Rwadiallahu Anhu’s time and has gone on ever since.
Secondly: Declaring (تداعى) during Taraweeh prayer is permtted. Meaning, Taraweeh is performed in Jamaat after declaration and is a Sunnate Muakkadaye Kifaya. And Tahajjud (تداعى) or declaring and performing Jamaat is Makruh Tahreemi and Bidyate Syeeyah, whether is is Ramadan month or any other month.
It has been mentioned that-
جماعت در تماز تهجد بتداعى مكروه است وبغير تداعى كه اتفاقا ايك دو كس يا سه كس اقتدائى تهحد نمايند مكروه نيست- التداعى بان يقتدى اربعة بواحد كما فى الدرر كذا فى الدرالمختار فاضل … درحاشيهء شرح وقاية مى ارد- الجماعة فى التهجد الذى هى نافلة بدعة قطعا. (مجموعهء فتاو .. سعديه)
Meaning:- Performing Tahajjud prayer with declaration is Makruh Tahreemi. And if coincidentially there are three people standing behind while performing Tahajjud prayer without declaration, it won’t be considered as Makruh (highly debated; some are for it and some are against it.) However if there are four people (agreed by all) it will be considered as Makruh. This has also been mentioned in Durar and Durrul Mukhtar. Allama Fazel Chalpi Rwahmatullahi Alalihi mentioned in Sharhe Bekayar Hashia, Tahajjud prayer which is a part of Nafl prayer is considered as Bidyate Syeeyah. (Majmuaye Fatwaye Swa’diah)
Therefore, it is proven that Taraweeh and Tahajjud are two different prayers. If they were the same, Taraweeh would’ve also been considered as Makruh Tahreemi and Bidyaati Syeeyah. Since according to Ulama-e-Kiram Tahajjud prayer will be considered as Makruh Tahreemi and Bidyaati Syeeyah if performed with Jamaat even if it’s Ramadaan. Mainly- Taraweeh and Tahajjud are very different prayers.