I perform Taraweeh prayers with Jamath in Masjid during Ishah. Can I pray Tahjud prayers later on at home during third part of night - before early morning? Is Taraweeh same as Tahjud or is it a different Salah.
It's not proven that Messenger of Allah (ﷺ) prayed Taraweeh and Tahajjud separately. I couldn't find anything which separates these prayers as different Taraweeh, Tahajjud, Qayam-ul-Lail, Atama
'A'isha, the wife of the Messenger of Allah (ﷺ), said that between the time when the Messenger of Allah (ﷺ) finished the 'Isha' prayer which is called 'Atama by the people, he used to pray eleven rak'ahs, uttering the salutation at the end of every two rak'ahs, and observing the Witr with a single one. And when the Mu'adhdhin had finished the call (for the) dawn prayer and he saw the dawn clearly and the Mu'adhdhin had come to him, he stood up and prayed two short rak'ahs. Then he lay down on his right side till the Mu'adhdhin came to him for lqama. (This hadith has been narrated with the same chain of transmitters by Ibn Shihab, but in it no mention has been made of Iqama )
Reference: Sahih Muslim, Hadith # 736 b-c Book 6
Narrated Abu Salama bin `Abdur Rahman:
that he asked Aisha "How was the prayer of Allah's Messenger (ﷺ) in Ramadan?" She replied, "He did not pray more than eleven rakat in Ramadan or in any other month. He used to pray four rakat ---- let alone their beauty and length----and then he would pray four ----let alone their beauty and length ---- and then he would pray three rakat (witr)." She added, "I asked, 'O Allah's Messenger (ﷺ)! Do you sleep before praying the witr?' He replied, 'O `Aisha! My eyes sleep but my heart does not sleep."
Reference:Sahih al-Bukhari 2013 Book 31, Hadith 6
First hadith tells that Messenger of Allah (ﷺ) only prayed 11 rakaat for the whole night after Isha which means he only prayed Tahajjud and in second Hadith Aisha (R.A) is again telling about Tahajjud but Ismael Bukhari REHMATULLAH put this hadith in
Kitab SALAH-TU-TARAWEEH which indicates that according to his understanding of Hadith the current hadith is talking about
SALAH-TU-TARAWEEH while it's clear that Aisha (R.A) is talking about Tahajjud
'Abdur Rahman bin 'Abdul Qari said,
"I went out in the company of 'Umar bin Al-Khattab one night in Ramadan to the mosque and found the people praying in different groups. A man praying alone or a man praying with a little group behind him. So, 'Umar said, 'In my opinion I would better collect these (people) under the leadership of one Qari (Reciter) (i.e. let them pray in congregation!)'. So, he made up his mind to congregate them behind Ubai bin Ka'b. Then on another night I went again in his company and the people were praying behind their reciter. On that, 'Umar remarked, 'What an excellent Bid'a (i.e. innovation in religion) this is; but the prayer which they do not perform, but sleep at its time is better than the one they are offering.' He meant the prayer in the last part of the night. (In those days) people used to pray in the early part of the night."
Sahih al-Bukhari 2010 Book 31, Hadith 3
According to Umar R.A the Salah of Taraweeh should be prayed at the last part of night which is the exact time when Messenger of Allah (ﷺ) prayed Tahajjud
These things all indicate that Taraweeh and Tahajjud are same - and according to Prophet Muhammad (ﷺ) this salah is better to be prayed at last part of night and the Salah should be as long as you can according to
let alone their beauty and length
ALLAH knows best