Vaishnavism and Buddhism
The analysis is done based on history centric opinions of western and Indian scholars. Also Swami Vivekananda has pointed many stuffs in this angle. In fact Swami Vivekananda is the first scholar to study Hinduism from a history centric point of view. It may sound different to mythology oriented people, but this is the true fact about ancient India.
To make long story short - we need to understand in details about three evolutions inside Hinduism - Brahminism, Tantra, Vaishnava.
Brahminism was an orthodox, conservative system, did not allow others to initiate into Vedic study. So others advent Tantra, a liberal system, but secret - worship of various deities through formal rituals - Shiva, Shakti, Ganesha etc are deities of Tantra. Vaisnava got developed later based on Purana, more liberal, and open in public - Vishnu as authority. And all these Puarans are written after Buddhism downfall. Shiva worship through Shiva-linga was present even before Buddhism. Buddhism had tried to dominate that Shiva worship. So they had built Buddhism monastery near to each Shaiva-peeta. e.g Buddha-gaya.
We need to analyse these in a history centric way. Our tendency is inclined towards mythology centric study based on Puranas. But we need to ask question - what is the origin of Purana?
A section of Buddhism known as Hinayana, were involved in idol worship. They were worshipping Buddha as God in temples, though Buddhism core belief system does not accept God. In fact they had built many temples in India. In Vedic Hinduism, there was no mention of temples, only Asrama concept was there.
Those Buddhists were later merged into Hinduism as Vaishnava and they only declared Buddha as Jagannatha or Vishnu. In fact there was no Buddhism in India. That itself was considered as Hinduism because all Hindu people were involved in worshipping Buddha. They borrowed vegetarianism from Buddhism. Otherwise in Vedic era, people used to eat meat. Link - http://www.srimatham.com/uploads/5/5/4/9/5549439/beef_in_ancient_india.pdf
Those worshipper Buddhists-cum-Vaishnavas wrote many Puranas. Those authors were called Suta-muni. Suta means lower caste people. For example Jagannatha temple in Orissa was a Buddhism temple, later converted into Vishnu. Link -
All this transition had happened over a five hundred years starting from 200 BCE. Later Buddhist worshipper groups formally separated from core Buddhism because they don't believe in God. And that was known as Vaisnava sect, worshipping Buddha as latest incarnation of Vishnu in various temples. Buddha is considered as one incarnation out of ten incarnation of Vishnu [Dasavatara]. When Buddhism was eradicated from India by Kumaril Bhatta, Shankara etc, they did not take Buddha name out of fear, but continued as Vishnu. In fact current Vaishnavism is denying any link to historical Buddha of 600 BCE. Rather it is replaced by a mythological Buddha.
Of late in 10th century, Ramanuja took birth in that Vaisnava group. Then Madhva, Chaitanya Dev etc gave more maturity to Vaisnavism. Now Vaisnavism is considered as a main stream faith system in Hinduism.