Wa `alaykum As-Salamu wa Rahmatullahi wa Barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
Dear sister in Islam, we commend your eagerness to become well acquainted with the teachings of Islam, which is the way Allah has chosen for the welfare of His servants.
As regards your question, it is true that the Qiyam and Tahajjud Prayers are the same prayer with different names. Tahajjud and Qiyam are the same thing, namely optional night prayer. However, the only difference appears when these two names are used during Ramadan. The prayer after `Isha' Prayer is referred to as Qiyam (or, more commonly, Tarawih), while spending the night in worship (after some sleep) is called Tahajjud. Both names have been used in the Qur'an. Referring to Qiyam, the Qur’an uses the phrase qum al-layl when Almighty Allah says: "O thou wrapped up in thy raiment! Keep vigil the night long (qum al-layl), save a little, a half thereof, or abate a little thereof, or add (a little) thereto and chant the Qur’an in measure…" (Al-Muzzammil: 1-4)
Using the verb fatahajjad, a derivative from the root hajada (meaning to keep vigil), Allah says: "And some part of the night awake for it (fatahajjad), a largess for thee. It may be that thy Lord will raise thee to a praised estate." (Al-Isra': 79) This last verse means to get up and perform Tahajjud Prayer. Tahajjud refers to getting up after sleeping and performing optional night prayer. Since this optional night prayer is most probably performed after sleeping for some time, it is named Tahajjud, though the original name of night prayer is Qiyam, which literally means standing. It is also said that Tahajjud is prayer after sleeping, while Qiyam refers to spending the night in worship in general, whether it is Prayer (salah) or any other form of worship.
For a comprehensive fatwa on optional night prayer, please read:
Tahajjud: Virtues and Way of Performance
