Buddhism is not a branch of Hinduism

"Hinduism is not a religion in the usual sense of the word, but is rather a collection of sometimes widely divergent religious concepts and practices that evolved out of Brahmanism in India. It is sometimes said that Buddhism is a branch of Hinduism or that it started as a reform movement within Hinduism. Neither of these claims is correct. Hinduism did not exist at the time of the Buddha and only began to evolve after the 3rd or 4th centuries CE. But even if Hinduism is confused with Brahmanism it is still clear from the Buddhist scriptures that the Buddha saw his Dhamma as contrasting with and being an alternative to the religion of his time, not a reform or a reinterpretation of it.

During the centuries in which they existed together, Buddhist and Hindu scholars and philosophers were sometimes highly critical of each others' ideas and practices. This would not have happened if the two were the same or similar. The Maitri Upaniùad says: `There are those who love to distract believers in the Vedas by the jugglery of false arguments, comparisons and parallelisms. The world is bewildered by a doctrine that denies the self (nairàtmyavàda), by false comparisons and proofs, it does not discern the difference between the wisdom of the Vedas and other knowledge ....... These and the many other criticism of Buddhism in ancient Hindu literature would have been unnecessary and meaningless if one was simply a reform of the other or the two were the same.

However, despite disagreements on the philosophical level, the relations between Buddhists and Hindus in ancient India were generally harmonious as both tend to be tolerant religions. "
~ Ven S. Dhammika.