Shesha Naaga and Vaasuki Naaga are brothers. Their mother is Kadru and father is Kashyap rishi. Kashyap rishi got married to 13 daughters of Daksha Prajapti. Amomg them Kadru and Vinata were asked for a boon. Kadru asked to be the mother of 1000 snakes and Vinata asked to be the mother of two sons but both of her sons should be brave than 1000 snakes child of Kadru.
When the time came Kadru gave birth to 1000 snakes including Shesha Naaga, Vaasuki, Takshak and Manasa where as Vinata gave birth to Garuda and Aruna.
Shesha Naaga was the bravest among the sons of Kadru. Shesha Naaga is also called as Anant. When Shesha Naaga saw his mother and brothers cheatng on Vinata, he left the family and went to place called Gandhmaadan to worship Brahma. He pleased Brahma. Brahma gave him a boon that his knowledge should never be separated from Dharma. Brahma also said that you should keep this Earth on your head and make it stable as it rotates always and Shesha Naaga followed the instructions. Lord Vishnu sleeps on Shesha Naaga in Sesha Sagar. According to scriptures the younger brother of Lord Rama, Lord Laxmana and elder brother of Lord Shree Krishna, Lord Balaram were the incarnation of Shesha Naaga.
Vaasuki Naaga is believed to be the king of snakes according to scriptures. Saat Sirsha is wife of Vaasuki Naaga. Vaasuki is also very religious and respects dharma. One day when Kadru cursed them ( Vashm hone ka Shraap) in a Sarpa Yagya Vaasuki was tensed. A Naaga named Ailapatra told Vaasuki that only the son from Vaasuki’s sister and Jarat Kaaru can overcome that curse. So after hearing that Vaasuki arranged the marriage. Vaasuki’s sister and Jarat Kaaru rishi got married. After sometime a son named Aashtik was born and he stopped that Sarpa Yagya and help to overcome everyone from that curse. Vaasuki was also used in Samundra Manthan. At the time of Tirpurdaha Vaasuki was used as the rope of Shiva Dhanush.
It is very hard to say who is better between these two. Shesha Naaga is also called as Aadhi Shesh where Aadhi means the first of all and Shesh means who still remains after an era. It is believed that after any era when Shesha Naaga opens his mouth Pingle, Rudra, Kapali, Virupaksha and Bheema come out and destroy everything. It is also said that Shesha Naaga have 1000 heads.
But there is no description about the head of Vaasuki Naaga in scripture. Shesha Naaga is protector of Lord Vishnu whereas Vaasuki Naaga was saved from Garuda by Mahadev himself. Shesha Naaga is called Anant in many Puranas, Lord Krishna in Bhagawat Geeta addresses about it saying Shesha Naaga is the Lord Vishnu of snakes. Which means Shesha Naaga is the Avatar of God. Vaasuki Naaga is younger brother of Shesha Naaga and he is only powerful after Shesha Naaga. Vaasuki is the necklace of Mahadev. Shesha Naaga left his family leaving the King’s throne empty to serve Lord Vishnu. After that Vaasuki got that throne and ruled for many years. Vaasuki also went to serve Mahadev which gave him the title ‘King Of Naaga’. After Vaasuki the throne was given to Takshak Naaga.
Both Shesha Naaga and Vaasuki Naaga have healthy relation between them in the scripture. Both of them are slaves of Tirdevs( Lord Vishnu And Mahadev) and there is no proofs of fight between them, so its hard to say who is better. But if we talk about strength and power Shesha Naaga is better than any other snakes.