Based on Lipner, J. (2006). The Rise of “Hinduism” ;or, How to invent a World Religion with only moderate success. Hindu Studies, 10(1), 91-104. Williams, P., & Tribe, A. (2000). Buddhist thought: A complete introduction to the Indian Tradition. London: Routledge. Read pp. 1-72. Question: Consider the basic tenets of Hinduism and Buddhism—how do they compare and contrast? 1. What impact did they have on Indian culture and society 2. Why did Buddhism eventually disappear from India?

According to Lipner, the primary difference between Hinduism and Buddhism is that Hinduism is an abstract cultural concept built around deity worship, while Buddhism represents a tighter philosophical narrative that focuses on the truth within. Another major difference is that, unlike Buddhism, Hinduism does not have an official founder and began as a rather nebulous cultural idea that was eventually codified into a religion for the sake of identifying its followers. Buddhism can be traced back to a specific founder and an ideology that began in India, while Hinduism sprang from Indian culture itself.
Williams and Tribe further illustrate the differences between Buddhism and Hinduism by explaining that the Hindu gods are more solid structures than the abstract concepts found within Buddhism. While Hinduism is characterized by a set of general principles and qualifications met by its followers, it features deities with distinct personalities that can be worshiped. Buddhism, while its principles are more clearly defined, does not require worship of any particular deity. In this sense, Buddhism and Hinduism are both compatible and distinct. A Buddhist can easily worship the Hindu deities without ideological conflict as Buddhism only requires its followers to pursue and practice Dhamma, or the continual journey towards salvation. Hinduism also emphasizes a form of salvation through religious practice, but Buddhism leaves room for followers to obtain salvation through Hinduism, Christianity or any other compatible religion.
1. What impact did they have on Indian culture and society
Hindu's greatest impact on Indian culture was to provide an umbrella under which existing Indian metaphysical and cultural practices and identities could live. The open nature of the Hindu religion both reflects and shapes the openness of Indian culture towards other cultural influences, encouraging the practice of other religions. Lipner theorizes that in the future, Hinduism may further the spread of globalization within India and lead to a polycentric approach to other world faiths.
Buddhism, as described by Williams and Tribe, has significantly influenced Indian culture and society by providing the Buddha as a spiritual teacher and an exemplar of enlightenment. With the introduction of Buddhism to India, many Indians came to consider themselves as polycentric, or open to both Buddhist and Hindu teachings. Buddhism also introduced the concept of Dharma, or ultimate truth, which made the search for truth and especially the truth within a significant cultural and religious value among Indians. In this sense, Buddhism is largely responsible for the emphasis on personal responsibility that is found throughout Indian culture and social values.
2. Although Buddhism was founded in India, Williams and Tribe explain that it had all but disappeared in India by the thirteenth century CE. Many Buddhists believe that this is part of the natural process, by which Buddhism will one day cease to exist in the world as a whole until it is ultimately rediscovered by a new Buddha. Nonetheless, while Buddhism faded in popularity in India, it remains prevalent in Japan, China, Tibet and Southeast Asia. Another reason why Buddhism largely disappeared from India was the increasing popularity of tantric Buddhism, which naturally led the faith to merge with increasingly popular religions, such as Jainism and Hinduism.
http://www.ahandfulofleaves.org/documents/Buddhist%20Thought_Williams_2000.pdf

https://www.jmu.edu/gandhicenter/wm_library/juliuslipnerlecture.pdf