What is the difference between night vigil prayer and tahajjud praeyr?

Night starts at sunset and continues until true dawn. Qiyamul-Layl (night vigil prayer is only performed after ‘Isha prayers.

Difference between Qiyamul-Layl and Tahajjud
1- - Qiyamul-Layl is spending the night, or even a brief period of it, praying or performing another act of worship. It can be performed either before or after sleeping. Qiyamul-Layl does not necessarily have to be comprised prayers or that it takes up the majority of the night. Additionally, it is not a condition to be preceded by sleep. Whoever spends the night or a brief portion of it reading the Qur`an, or listening to hadiths or glorification of Allah, or in invoking peace and blessings upon the Prophet, whether before or after he has slept for a portion of the night, is considered to have performed Qiyamul-Layl.

2-- Tahajjud is one of the forms of Qiyamul-Layl. It is performed by praying at night after sleeping. It is specific to prayers that are preceded by sleep. If Tahajjud is not preceded by sleep and does not comprise prayers, it is not considered Tahajjud but Qiyamul-Layl. Tahajjud is the optimal form of Qiyamul-Layl. This is because the Prophet (pbuh) used to perform Tahajjud. Allah the Almighty says, “And during the night, wake up and pray, as an extra offering of your own” [17:79).

The best time to perform Tahajjud is the latter part of the depths of the night. This is due to the report of Amr Ibn Absa who said, “I said, ‘O Messenger of Allah! During which part of the night are prayers most likely to be answered?’ he replied, ‘The middle of the latter portion of the night, so pray at that time with whatever you wish’” [recorded by Abu Dawud and Al-Tirmidhi).

If the night is divided into six parts, then the fourth and fifth parts are the best for prayer due to the report of Abdullah Ibn Amr (may Allah be pleased with them both) who said, “The Messenger of Allah (pbuh) said, ‘The most beloved prayer to Allah is the prayer of Dawud (peace be upon him) and the most beloved to Him is the fast of Dawud—he used to sleep half of the night, spend one third of it in Qiyamul-Layl, and then sleep for one sixth of the night. He used to fast every other day” [recorded by Bukhari in his Sahih).

If one wishes to divide the night into two parts by sleeping during the first half and spending the other in Qiyam, then the second half is best. This is because during this time, it is less likely for one to commit sins and because it includes the last third of the night. Abu Huraira (may Allah be pleased with him) said, “The Messenger of Allah said, ‘Allah descends every night to the lowest heaven and say, “Who is supplicating me that I may answer him? Who is asking me [for something] so that I may give him? Who is asking for My forgiveness so that I may forgive him?” [recorded by Bukhari and Muslim).

Number of rak’as (prayer cycles) in Tahajjud prayers
Scholars are unanimous that the minimum number of rak’as is two short ones. This is based on the report of Abu Huraira (may Allah be pleased with him) who said, “If you get up during the night, start your prayer with two brief rak’as” [recorded by Muslim in his Sahih]. There is no maximum limit to the number of rak’as that may be offered. It was reported that the Prophet (pbuh) performed 13 rak’as of Tahajjud and one rak’a of Witr. There is no harm in offering either more or less than the number of ra’kas the Prophet offered.