I have heard that there are three subschools of Advaitha Vedanta.
Are there more sub schools?
Can you briefly summarize the main philosophical differences in the schools?
The idea that these are hard and fast 'schools' can be misleading. These are all highly diacritical streams of logical argumentation of the Advaita as taught by Adi Sankara. Many of the ideas in each are the same. Three of Sankara's disciples were Padmapada (who wrote the Pancha-padika and whose disciple Prakashatma wrote the Vivarana commentary), Sureshvara (Vartika), and Vachaspati Mishra (Bhamati commentary). Although scholars and others often refer to them as schools, it is best to understand that all three draw upon Shankara and his writings. Most of the 'differences' are specious arguments; all three use Shankara as reference. Except for the sake of pure argumentation, does it matter whether Advidya arises from a cosmic standpoint or an individual standpoint? Avidya is Avidya. From the standpoint of Brahman both viewpoints are non-existent. Sharma, in his book The Advaita Tradition in Indian Philosophy: A Study of Advaita in Buddhism, Vedanta, and Kashmira Shaivism, says p 222:
...But really speaking, the two schools [Vivarana and Bhamati] are not opposed. Both the schools agree that Isvara is ever free from ignorance and that in Him sattva predominates. Hence, whether concealment and projection are treated as as two aspects of the same power or the former is called avidya and the latter maya, the difference is only in words.
Sharma, quoting Shankara, says (A Critical Survey of Indian Philosophy, p 288 - https://archive.org/details/IndianPhilosophyACriticalSurvey):
...Shankara remarks that reasoning, because it depends upon individuals, has no solid foundation. Arguments held valid by some, may be proved fallacious by others more ingenious. ( Shariraka-Bhasya, II, I, 11)
The commentators after the above mentioned 3 'schools', focused on the Vivarana and Bhamati streams. Some of their differences are dependent upon Shankara's similes and metaphors of the rope/snake, universal space/space in a jar, shell/silver, water/waves, etc, etc. It should be remembered that these are metaphors. Sharma cautions (p 228, Advaita Philosophy in Indian Philosophy):
...are suggested for their value and should not be taken literally and woven into theories...
and p 229:
Really speaking, the distinction among Ishvara, jiva, and saksi is valid only in empirical life; ultimately all the three are Brahma itself. Ishvara, jiva and jagat are appearances projected by maya, they arise together and they vanish together when maya is canceled.
Swami Vivekananda says (Complete Works, V7, pp164-5):
For this reason, nescience [avidya, or maya] has no absolute reality. You can call nescience neither real nor unreal; "Neither real, nor unreal, nor a mixture of both". About a thing that is thus proved false, neither question nor answer is of any significance. Moreover, any question on such a thing is unreasonable.
Another book on the same subject is Bhamati and Vivarana Schools of Advaita Vedanta: A Critical Approach by P.S. Roodurmum usually available here - https://www.mlbd.in/
After going through lot of research i finally got the results in my hand i am putting references on the last because Advaita Vedanta is very vast to study.
"And I have to say this is a very brilliant question"
Advaita Vedanta is believed to be one of the classic paths to spiritual realization in Hindu tradition.
Advaita Vedanta is the oldest extant sub-school of Vedanta
There are only three sub schools of Advaita Vedanta:
let’s see in general what are the three sub-schools of Advaita Vedanta.
Bhamathi sub school bases itself on the Bhamathi sub commentary of Vachaspathi Mishra on Sankara’s Brahma Sutra Bhashya. Scholars opine that there was an advaitin Mandana Misra who was a contemporary of Sankara and wrote the work Brahma Siddhi. The means propounded by this Mandana Mishra in his Brahma Siddhi (independent work) is different from Sankara and traditional Advaita Vedanta. Thus there are differences of opinion on this (many places Sureshwara too refutes the views of Mandana Mishra in his Naishkarmya Siddhi) (Sankara’s and Mandana Mishra’s means to the final goal of Advaita). Vachaspathi Mishra wrote the first complete commentary on Sankara’s BSB titled Bhamathi (there was Panchapaadika of Padmapada acharya but this work analyzes only the first four sutra bhashyas (but there are proofs from Bhamathi that Padmapada had written an entire commentary on Sankara’s BSB and Madhava vidyaranya’s Sankara dig vijaya too tells it the same way – but scholars differ on this). Scholars thus opine that Vachaspathi tried to harmonize Sankara with Mandana Mishra – thus in many places, he uses Brahma Siddhi means which might seem to be against Sankara’s means. Even in some places, vachaspathi goes against Sankara’s bhashya to interpret in a different way which is supported in Amalananda’s Kalpatharu sub-commentary on Bhamathi (it is interesting to note that Prakataartha Vivarana of Anubhooti svaroopacharya refutes the places where bhamathi goes against Sankara - but the views of this acharya is refuted and bhamathi supported by Amalananda). It finally ends up in Appayya Dikshitar’s parimala sub commentary on Kalpatharu where dikshitar supports Sankara while refuting Bhamathi in those places. Enough of this too much explanations ---- this sub school is based on Bhamathi mainly and Kalpatharu-Parimala secondarily to interpret Bhamathi properly. This is the first sub-school of Advaita Vedanta. There are less number of followers for Bhamathi school these days (most followers are in north I believe) but still it is interesting to note that Gauda Brahmananda Saraswathi in his Nyaaya Ratnaavalee on Siddhanta Bindu of Madhusudana Saraswathi says that Vedanta shaastra includes the five works of Brahma Sutra, Sankara’s Brahma Sutra Bhashya, Bhamathi, Kalpatharu and Parimala. It is generally held that there are very few who have read all the five works and mastered them (the entire five works are published by Nag Publishers in two volumes for 400 Rs. Chaukhambha Press too have brought out the five works for 650 Rs.).
As mentioned earlier, Padmapaada had written Panchapadika on Sankara’s Brahma Sutra Bhashya which was again explained in detail by Prakaasaatman in Panchapaadika Vivarana. There are many sub commentaries on Vivarana of Prakaashaatman. The school which bases itself on Vivarana of Prakaasaatman is called Vivarana school. The important sub commentaries on Vivarana are Akhandaananda Tattva Dipana and RjuVivarana of Sarvajna Vishnu – there are as well sub commentaries on it by Chitsukha and Nrsimha ashrama. Amalananda has also written Panchapaadika Darpana. This school is very famous and has many acharyas following it. Two of the three complete commentary on Sankara’s BSB (one being bhamathi) follow Vivarana school only – these are Anandagiri’s Nyaaya Nirnaya and Govindananda’s Bhashya Ratna Prabha. We can very clearly say that this school bases on one of Sankara’s direct shishya or disciple (Sachidanandendra Saraswathi does show in some work that this school differs from Sankara’s BSB but it is worthy enough to analyze on that – might be we can take that up for Ph. D when lot of free time is there).
Sureshwaracharya, after being guided by Sankara, wrote vartikas on Sankara’s Taittiriya and Brihadaranyaka Bhashyas. Vartika is that commentary in which ukta, anuktha and duruktas are mentioned – ukta is explaining in detail whatever the original work does – anuktha is explaining that which has been left out in the origin – duruktha is making correct whatever has been explained either wrongly/incompletely in the original work. Sureshwaracharya thus differs in almost 4-5 places in the Brihadaranyaka Vartika from Sankara’s Brihadaranyaka Bhashya. The Brihadaranyaka Vartika consists of as much as 12,000 slokas which has been further commented by Anandagiri and vidyaasagara. Vidyaranya has condensed the vartika into a work called Vartika saara. The school based on Vidyaranya’s as well as Bharathi Teertha’s works (which themselves base on Vartika of Sureshwara) is called Vartika school. We can thus very well clearly find differences between ishwara in vivarana and this school. Vivarana considers ishwara as Bimba Chaitanya (that original consciousness which is reflected and becomes jeevas) whereas vartika school considers both ishwara and jeeva as illusory and not real at all (this is clearly found in Panchadashi many places). But we have to remember that Vidyaranya and Bharathi Teertha have written vivarana prameya sangraha which is summarizing Vivarana of Prakaasatman where they follow the vivarana school very clearly. I have also found in one place in panchadashi where vidyaranya refutes bhamathi view of jeeva.
advaita vedanta : https://en.wikipedia.org/wiki/Advaita_Vedanta
sub-schools of advaita : http://jeevaaspervarioussystems.blogspot.com/2006/09/different-sub-schools-of-advaita.html