All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
We have already issued Fatwa 83333 about the ruling and method of performing the Tahajjud (i.e. optional night prayers), so please refer to it.
We have also published Fatwa 88878 regarding performing the optional prayer quietly or loudly, so please refer to it.
As regards your question about the difference between Tahajjud and Witr, then some jurists define Tahajjud as an optional night prayer that a Muslim performs by night after sleeping at the beginning of the night; some other jurists define it as an optional night prayer after 'Ishaa prayer even if this is before going to bed. According to the latter definition, Tahajjud is equal to Qiyaam-ul-Lail, but according to the first definition Tahajjud is more specific than Qiyaam.
As regards the Witr prayer, it is the odd number of Rak'ahs with which one concludes his prayers at night whether after Qiyaam or after 'Ishaa.
The least number of Rak'ahs to be considered Witr is one single Rak'ah whereas the least rank of perfectness is to pray three Rak'ahs for Witr. This means that the Witr prayer is included in Tahajjud so it is desirable for a Muslim who performs Tahajjud at night to finish his night prayers with Witr no matter how many Rak'ahs he had prayed. Ibn 'Umar reported that the Prophet said: "Make the Witr prayer the last of your prayer at night." [Al-Bukhari and Muslim]
According to the view of the majority of scholars the Witr prayer is a confirmed Sunnah while some others consider it to be obligatory, so a Muslim should be keen on praying it.
Allaah Knows best.