I can see both the prayers are prayed in my Masjid (mosque).
Taraweeh time 8:45 to 10:00 pm Qiyaam al-layl time 2:30 to 3:30 am
What is the difference between Taraweeh prayer and Qiyaam al-layl prayer?
I need some authentic references over this.
Qiyam al Layel is any prayer from Isha until Fajir whether it is in Ramadan or in any month of the year. It is not restricted between 2:30 to 3:30. But I think I understand what you mean. Mosques announce Qiyam in Ramadan at this time for some reasons:
1- It is the most sacred time in which Allah descends to the lowest heaven.
It was narrated by al-Bukhaari in his Saheeh (1145) and by Muslim (1261) from Abu Hurayrah (may Allaah be pleased with him), that the Prophet (peace and blessings of Allah be upon him) said: "The Lord descends every night to the lowest heaven when one-third of the night remains and says: ‘Who will call upon Me, that I may answer Him? Who will ask of Me, that I may give him? Who will seek My forgiveness, that I may forgive him?
2- It is the most suitable time for Muslims to pray at. It gives them enough time to rest since it starts at 2:30 and gives them enough time to go back and eat before fajir athan since it ends at 3:30.
Hence, Qiyam can be prayed at anytime, anywhere, with or without people.
Tarwaeeh is considered Qiyam al Layel. However, Scholars called it Taraweeh to distinguish it from other prayers since it is done in Ramadan, just after Sunnah of Isha, and not lengthy one.
In the hadith:
Prophet (PBUH) led his companions in prayer (Taraweeh) for several nights, then on the third or fourth night he did not come out to them. When morning came he said: "Nothing prevented me from coming out to you except the fact that I feared that it would be made obligatory for you."
Narrated by al-Bukhaari, 1129. According to the version narrated by Muslim (761): "But I feared that night prayers would be made obligatory for you and you would not be able to do them."
To understand why Umar asked Muslims to pray Taraweeh after the death of the prophet Muhammed (PBUH) read this Why Umar (r.a) said praying taraweeh in jamaah was "good bidah"?