Al-jawab billahi at-taufeeq (the answer with Allah's guidance)
1) Praying Tarawih Salah with congregation has more reward than praying it alone.
That he was informed by 'Aisha (RadhiyAllahu Anha), "Allah's Apostle went out in the middle of the night and prayed in the Masjid and some men prayed behind him. In the morning, the people spoke about it and then a large number of them gathered and prayed behind him (on the second night). In the next morning the people again talked about it and on the third night the Masjid was full with a large number of people. Allah's Apostle came out and the people prayed behind him. On the fourth night the Mosque was overwhelmed with people and could not accommodate them, but the Prophet came out (only) for the morning prayer. When the morning prayer was finished he recited Tashah-hud and (addressing the people) said, "Amma ba'du, your presence was not hidden from me but I was afraid lest the night prayer (Qiyam) should be enjoined on you and you might not be able to carry it on." So, Allah's Apostle died and the situation remained like that (i.e. people prayed individually). "
(Bukhari, Book 32, 3:229)
"It is narrated that one night during Ramadan Rasulullah (Sallallahu Alaihi Wasallam) performed Salaat al-Tarawih in the masjid. A group of Sahaabah joined him during his salaah. The following night the same happened as the previous night except that the number of followers had increased considerably. Hence on the third (or fourth) night Rasulullah (Sallallahu Alaihi Wasallam) did not come out to the Masjid to perform Salaat Al-Tarawih with the people. The following morning he said to them: "Indeed I had seen your eagerness (to perform the Tarawih behind me), but for the fear that this Salaah will be made Fardh (compulsory) upon you during Ramadan, I did not come out to join you in the Tarawih". (Muslim, Hadith no.761)
Prophet (Sallallahu Alaihi Wasallam) prayed 20 Rakat, narrated by Ibne Abbas (RadhiyAllahu Anhu): that during the month of Ramadan, Prophet (Sallallahu Alaihi Wasallam) performed 20 Rakat and thereafter, performed the Witr Salat (prayer). (Baihaqi and Musannaf Ibne Abi ShaibahBA) Hafiz Ibne Hajar states this Hadith in his book, (Al Mataalibul Aaliyah Vol.1 Pg.146).
Please read more: http://www.muftisays.com/qa.php?viewpage=viewQA&question=2622
From the above mentioned Ahadith we understand that in the beginning the Tarawih Salah with congregation in the Masjid was not encouraged due to the fear that this Salah would become compulsory. When the Messenger of Allah (Sallallahu Alaihi Wasallam) died the fear for it to become compulsory was removed and thus became a virtuous and preferred act to pray with Jama'ah in the Masjid. (See: Bazlul Majhud Vol.2 Pg.336)
What does Qiyam Ramadhan refer to?
According to Allamah Aini and Hafiz Ibn Hajr (RahimahumAllah) Qiyam Ramadhan is 'Aam (general) i.e. it refers to any Nawafil Salah (incl. Tarawih), and praying them (Nawafil) in Jama'ah is permissible. (Fathul Bari Vol.4 Pg.251)
In a narration Nabi (Sallallahu Alaihi Wasallam) said, "Whosoever stand in prayer in Ramadhan with conviction and hoping for reward will have his pasts sins forgiven." (Bukhari, Muslim)
The above mentioned Hadith, "stand in prayer in Ramadhan" refers to any Salah and not specifically to Tarawih, and whosoever performs any Salah during the night of Ramadhan will be rewarded.
However, there is no mentioning of Jama'ah (praying the Nawafil with congregation).
In brief, the places where Allamah Aini makes mention of Qiyam Ramadhan as 'Aam (general) he does not mention the issue of Jama'ah (congregation) and the places where the issue of Jama'ah is concerned he has not mentioned Qiyam Ramadhan but instead the word Tarawih is used like, "the Tarawih itself is Sunnah and to perform it with Jama'ah is Mustahab." Here he has not mentioned Qiyam but Tarawih. ('Umadatul qari Vol.5 Pg.267)
According to Allamah Nawawi and Kirmani and the general Fuqaha and Muhadditheen (Alaihimur Rahamah) Qiyam Ramadhan is referred specifically to Tarawih Salah.
(Fathul Bari Vol.4 Pg.251)
It is narrated by Salman Farsi (RadhiyAllahu Anhu) that he says, "Rasulullah (Sallallahu Alaihi Wasallam) addressed us on the last day of Sha'ban, relating "O people! verily a great month has shaded you, a blessed month containing a night better than a thousand months. Allah Ta'ala has made fasting compulsory in this month and has made the act of standing in prayer during the night as an optional act..." (Baihaqi)
The above mentioned Hadith, "the act of standing in prayer during the night as an optional act.." refers to Tarawih and not Tahajjud, because Tahajjud is an optional act performed in Ramadhan and besides Ramadhan which is not a speciality of Ramadhan. However, Tarawih is a speciality of Ramadhan because Tarawih is only performed in Ramadhan. Therefore, Qiyam here means Tarawih.
In addition, the word Qiyam (i.e. Tarawih) of this Hadith is supported by Fuqaha affirming it, this is mentioned in Fathul Qadeer Vol.1 Pg.333, Fatawaa Bazzaziyah Vol.4 Pg.29, Tahtawi Pg.234.
It is reported by Saib Ibn Yazeed, that the people used to do Qiyam (i.e. perform 20 Rakat of Tarawih) in the month of Ramadan during the era of Umar (RadhiyAllahu Anhu).
(Baihaqi As Sunanul Kubra Vol.1 Pg.496- Declared authentic by Nawawi, Iraqi and Suyuti).
Also, commentators of Hadith mentions Qiyam as Tarawih Like, Imam Muslim (Rahmatullahi Alaih) in Muslim, chapter: Encouragement to stand in prayer in Ramadhan i.e. Tarawih - Vol.1 Pg.259.
(See also: Al Arfush Shazzi Vol.1 Pg.329, Al Kawkab Ad Durri Vol. 1 Pg.267)
In Fatawaa Qadhi Khan, "It is Mustahab and virtuous to perform the Tarawih prayers with Jama'ah in the Masjid." (Fatawaa Qadhi Khan Vol.1 Pg.213)
The 20 Rakat of Tarawih with Jama'ah is the Sunnah of the Prophet (Sallallahu Alaihi Wasallam) and the consensus of the Sahabah (Companions) (Ijma-as-Sahaba). (al-Mughni vol.1, pg 803)
However, there are no established evidences from the Prophet (Sallallahu Alaihi wasallam) and the Sahabah (RadhiyAllahu Anhum Ajma'een) on any Nafl Salah with a large congregation. There is a Hadith reported by Ibn Abbas (RadhiyAllahu Anhu) that the Prophet (Sallallahu Alaihi Wasallam) was praying Tahajjud Salah and I stood up by his left side but he made me stand to his right.."
Here the Prophet (Sallallahu Alaihi Wasallam) led only Ibn `Abbas in congregational Tahajjud Prayer once. This is distinguished from Tarawih, as Tarawih was performed by a large number of people many times.
2) Praying Tahajjud Salah alone has more reward than praying it with congregation.
To pray a Nafl Salah with congregation is Makruh Tahreemi (highly disliked), whether the Nafl prayers are performed in Ramadhan or besides Ramadhan, except Salat Tarawih, Istisqa (prayer of rain), Eidain (two Eids) and Kusuf (prayer of solar eclipse). This is the Maslak (path) of the general Fuqaha and Muhadditheen. According to Imam Shaf'i and Imam Ahmad Bin Hanbal (RahimahumAllah) it is permissible to pray the Nafl prayers in congregation.
It is disliked to pray Nafl prayers with a large congregation or to make it a habit, except the Tarawih prayer in Ramadhan. (Raddul Muhtar)
Ibn Humam (Rahmatullahi Alaih) says, "the Nafl prayers in congregation are disliked besides Qiyam Ramadhan i.e. Tarawih and Salatul Kusuf." (Fathul Qadeer Vol.1 Pg.438)
Therefore, the Jama'ah of Tahajjud and other Nawafil Salah, whether in Ramadhan or besides Ramadhan will be disliked.
(See also: Raddul Muhtar Vol.1 Pg.664, Fatawaa Bazzaziyyah Vol.4 Pg.31, fathul Qadeer Vol.1 Pg.438, Al Bahrur Raiq Vol.2 Pg.71, Al Arfuz Shazzi Vol.1 Pg.230, Fatawaa Rashidiyyah Pg.289, 307- Fatawaa Uthmani Vol.1 Pg.446-458)
And Only Allah Ta'ala Knows Best.