What is the difference between Thenkalai and Vadakalai sects of Sri Vaishnavism?
First of all, these aren't the two main sub-sects of Vaishnavism, they are the two main sub-sects of Sri Vaishnavism. There are other Vaishnava Sampradayas, like the Gaudiya Vaishnava sect that ISKCON adheres to. What distinguishes Sri Vaishnavas from other kinds of Vaishnavas are mainly two things: our beliefs, and our origins.
Gaudiya Vaishnavas believe in Achintya Bheda Abheda, a simultaneous unity and difference between Jivatma (the human soul) and Paramatma (divine soul or supersoul) that is inconceivable. Madhvas believe in Dvaita, a fundamental difference between Jivatma and Parmatma. Sri Vaishnavas, on the other hand, believe in Visistadvaita, or "qualified monism", where there is one big unified whole called Paramatma, but within that there are still distinct parts like Jivatmas, akin to how an unborn baby is part of the body of a pregnant woman, but the pregnant woman's body extends beyond the baby (what Western thinkers would call panentheism).
The Sri Vaishnava sect has its origins in the works of the Alwars, a group of 12 ancient Vaishnava saints who lived in Tamil Nadu and and are famous for their poetry in praise of Vishnu. The 4000 poems of the Alwars were collected by Nathamuni into a book called the Naalayira Divya Prabhandam, which started being thought of as the Dravida Veda, or Veda of South India. It is the beliefs and principles embodied in the poems of the Naalayira Divya Prabhandam that provided a basis for the Sri Vaishnava belief system. Nathamuni was the founder of the sect, but it really got into prominence when Ramanujacharya became its leader. As I describe in this answer, it is after the time of Ramanujacharya that Sri Vaishnavas became a sizable enough portion of the Brahmin community that they started developing a distinct identity, which is e.g. how the Iyengars emerged from the Iyers.
Now Ramanujacharya placed pretty much equal emphasis on both Sanskrit scripture, like the Vedas, Upanishads, and Puranas, and the Tamil Naalayira Divya Prabhandam, because he took the notion that it was the Dravida Veda very seriously. But after his death, his followers started having disagreements. They all agreed on the Visistadvaita philosophy that characterizes the Sri Vaishnava belief system, but they had disputes that fundamentally revolved around how much authority we should give to the Naalayira Divya Prabhandam. Manavala Mamunigal and his successors believed that words of the Alwars are of paramount importance, and that they contain all the important messages of the Vedas and more, so they started called themselves Thenkalai (the Southern group), because their supreme scripture was the Naalayira Divya Prabhanda, which is from South India. Vedanta Desikan and his successors still believed that Alwars' words were sacred, but they also thought that we shouldn't neglect the importance of Sanskrit scripture like the Vedas. Since they placed some emphasis on scripture that had come from North India, they became known as Vadakalai (the Northern group).
The other doctrinal differences between Thenkalais and Vadakalais emerge from this split over what scriptures to pay the most attention to. For instance, Naalayira Prabhandam consists in large part of devotional poems where the Alwars describe their utter dependence on Vishnu for salvation (Moksha), akin to a lovesick youth being dependent on the affections of his beloved. Since the Thenkalais paid attention almost exclusively to the works of the Alwars, they developed the notion that all of us are dependent on Vishnu to grant salvation to whomever he chooses, and that we are powerless to do anything to achieve it. The traditional analogy is to baby kittens, who don't need to do anything in order to be carried around by their mother. The Vadakalais, on the other hand, got from Sanskrit scripture the insight that there are actually things we can do to try to achieve Moksha, like the principles of Bhakti Yoga laid out in the Bhagavad Gita. But they also paid attention to the poems of the Alwars, which led them to believe that it is possible to be granted salvation by Vishnu the way the Alwars were talking about, but in order to do that you need to first make a positive act of surrender (Saranagati or Prapatti) to the lotus feet of Vishnu. The traditional analogy for the Vadakalai belief is baby monkeys, who have to make some effort to cling to their mother's body.
The "cats vs. monkeys" dispute is the biggest doctrinal difference between Thenkalais and Vadakalais, but there are other smaller differences; here is a list of ten of them (the central difference I discussed above is described in points 1, 4, and 5):
1) Regarding Lord's mercy. Next to the Caste mark, this probably is the only other difference most people are aware of
Vadakalai View: Some positive gesture is necessary on the part of the jeevatma to deserve the grace of God, because He can be deemed partial if He grants Moksha to all both deserving and undeserving.
Tenkalai View: Lord's grace is spontaneous. He can grant Moksha to anyone he likes.
2) Regarding the status of Lakshmi (i) as to her being the means (ii) as to her being infinite (iii) as to her being Paramatma
Vadakalai View: (i) She is the means for attaining salvation as much as the Lord Himself and also has the role of a mediator ( Purushakara) (ii) She is infinite in nature (Vibhu) like the Lord Himself (iii) She is also Paramatma as much as the Lord Himself
Tenkalai View: (i) Do not accept this position though they accept her recommendatory role as held by Vadakalais (ii) She is atomic in nature like other Jeevatmas (iii) She is a Jeevatma like any of us.
3) Regarding Kaivalya
Vadakalai View: (i) Kaivalya is inferior to Paramapada (ii) Kaivalya is not eternal (iii) Kaivalya is situated Outside Paramapada
Tenkalai View (i) Accepted (ii) Kaivalya is eternal (iii) Kaivalya is within Paramapada but in its outermost parts.
4) Regarding the means of Bhakti and Prapatti
Vadakalai View: Accept both as the direct means but Bhakti is more difficult and dilatory while Prapatti is easy and immediate
Tenkalai View: Do not accept any means because Jeevatma is so utterly dependent as to be incapable of adopting either Bhakti or Prapatti as a means.
5) Regarding Prapatti
Vadakalai View: Prapatti has to be a positive specific act of surrender by the jeevatma to the Paramatma
Tenkalai View: No positive, specific act is necessary. All that is required is (i) the knowledge of the Svarupa of the Jeevatma and (ii) mental acceptance of the Lord's grace in granting salvation
6) Regarding sins
Vadakalai View: When a jeeva surrenders, the Lord forgives the sins committed by the jeevatma and grants Moksha.
Tenkalai View: The sins of a jeevatma is a source of joy for the Lord who relishes the same like a cow licking off the dirt on the body of its calf
7) Regarding performance of Compulsory duties like Sandhyavandanam
Vadakalai View: As compulsory duties are laid down by the Sastras which are the Lord's commandments, non- performance will tantamount to transgression of His commands (Ajna adhilangana) and will render the Prapanna liable for punishment
Tenkalai View: To a highly evolved soul, non- performance of the compulsory duties is not an offence. But, they should continue to do them more for setting an example to the less evolved souls.
8) Regarding the interpretation of the words "Sarva Dharman Parityajya' occurring in the Charama sloka
Vadakalai View: The Dharmas actually refer to the 32 Vidyas attaching to Bhaktiyoga which had already been given up by the jeeva due to incapacity and delay involved in observing them and the Lord offers to stand in their place
Tenkalai View: This is literally interpreted to mean ' First, give up your duties and then take refuge in the Lord'
9) Regarding the Lord's grief at the suffering of the souls
Vadakalai View: One can have grief only when one cannot remove suffering of another. But, the Lord is capable of removing suffering. So, there is no need for Him to grieve. As Sri Rama , He shows to the World how a human would feel and how one should react on seeing the misery of others.
Tenkalai View: They hold that the Lord actually feels sorry on seeing the sufferings of souls and cite examples from Srimad Ramayana where Sri Rama is depicted as grieving over the misery of others.
10) Regarding the Lord's being also atomic as well as gigantic in size as mentioned in the Vedas.
Vadakalai View: He is smaller than the atom in beings that are atomic in size. This is called 'Antar Vyapti' ( Immanence). He is also greater than the greatest in the sense He pervades and surrounds everything. This is called ' Bahir Vyapti'. ( Transcendence)
Tenkalai View: His being atomic in atoms and enveloping even the biggest are all done by what is known as 'Agatitha Ghatana Saamartya'- Special powers enabling accomplishment of even the impossible.
Tell me if you want me to clarify any of these points. By the way, in the interest of full disclosure, my dad is a Thenakalai and my mom is a Vadakalai, so I'm technically a Thenkalai, but my beliefs lean more toward the Vadakalai side because I think Sanskrit scripture is important. So that might bias my description of the two sides.
EDIT: The web page I quoted above gives only 10 of the 18 points of difference between the beliefs of Thenkalais and Vadakalais, but here is the full list of 18 points, quoted from a journal article. (Note that N denotes Vadakalai and S denotes Thenkalai.). Also, here is an excerpt from C. Umakanthan's book "Greatness of Saranagati in Sri Vaishnavism" discussing seven of the differences. (Vedanta Desikan's view is what became the Vadakalai view and Pillai Lokacharya's view is what became the Thenkalai view).
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