Bismillah
Ok here I m, Dan see originally the word taraweeh is not used in Quran but rather the term Qiyam which is to stand up. In essence, the term is Qiyamu layal. It means to stand up during night for prayer.
where did we get this from? ok here is the ayah:
"Stand (to pray) all night, except a little" (Quran 73:2)
The Messenger salla Allah a`lyhee wa sallam performed the 20 Raka`a after Esha using different locations. I mean he varied between home and Masjed. But always individually. Note if he did perform it always at the Masjed in congregation, it would have been obligatory. But Allah The Merciful allowed us a chance to benefit of the reward using different manners.
But of course he performed them at the Masjed for people also to learn about it.
Why later on people started calling them taraweeh? because the word Taraweek is derived from Raha which is rest or break. People used to break every two sets, I mean every 4 raka`as.
As for the Tahajud:
"And in some parts of the night (also) offer the prayer with it (i.e. recite the Qur'an in the prayer), as an additional (night) prayer for you (O Muhammad). It may be that your Lord will raise you to a station of praise and glory (in Paradise!)." (Quran 17:79)
As u can see Allah Promised the Messenger salla Allah a`lyhee wa sallam a special high rank in jannah thru this type of prayer. But also note that it was in particular ordained for him. Therefore, copying him in this honorable act Insh a Allah raises us in ranks.