This is a very interesting discussion.
I am not well-read in Buddhism as I am in Hinduism, but I am familiar with some of the Buddhas teachings and some of the facts about Buddhism. Now, Buddhism itself does not refer to a singular philosophy or belief system, but to several, each with varying interpretations and their own idiosyncracies. The Zen schools of Buddhism for example are more of a syncretic form of Buddhism, combining Taoism with Buddhism. Some of these branches are closer to Hinduism than others, such as Tibetian Buddhism.
It would be safe to say that Buddhism like all Indian religions religions has sprung from the Vedas. However, later the Buddhism branched off from Hinduism and established itself as its own religion. A similar phenomenon has taken place with Sikhism. However, technically, they would be considered denominations of Hinduism as opposed to new religions, and many Hindus do consider them as such and it is not uncommon for Hindus to practice them.
One of the most fundamental disagreements between Hinduism and Buddhism is their philosophy on the nature of self. While the Hindus contend that the Atman is real, the Buddhists reject the self and contend anatman(no self) and this has been the reason for the rivalry between Hinduism and Buddhism for centuries. The Hindus accept that there is a permenant all pervading supreme consciousness/supreme being that is within each and everyone of us, but the Buddhists accept that there is no self, that it just an illusion that arises from an aggregate of mental and physical processes and is continuously born and reborn every moment - the only real truth is nothingness.
Hence, based on these positions Hindu and Buddhist logicians have been debating for centuries, but Buddhist logicians have never been able to beat the Hindu logicians. The argument is, if the self is simply an impermenant and unending aggregation of mental and physical processes, how does this explain the fact of memory and personal identity which are enduring phenomena?
Yet, within Buddhism itself, within the most authorative texts such as the Mahanirvana sutras, we read this:
When I have taught non-Self, fools uphold the teaching that there is no Self. The wise know that such is conventional speech, and they are free from doubts. "When I have taught that the tathagata-garbha is empty, fools meditatively cultivate [the notion] that it is extinction [uccheda], subject to destruction and imperfect. The wise know that it is [actually] unchanging, stable and eternal."
It would would seem that those who accept this text have barely any differences with Hinduism. Hinduism propounds exactly the same thing. In fact it would seem what has happened is that some Hindus and Buddhists have misunderstood what Buddha meant when he said there was no self. He was referring to the phenomenal self, the same phenomenal self which in Hinduism is also said to be false. The notion of 'I' or ego is as strongly repudiated in Hinduism as it is in Buddhism. Hinduism also holds that this 'I' or ego is an aggregate of mental and physical processes which is temporal and impermenant. Like Buddhism, Hinduism also holds that that the phenomenal universe is temporal and impermenant.
It appears Buddha decided to only concentrate on teaching this to people, rather than teaching them about Atman and Brahman, to destroy the false notions of the world they had. He knew that by taking the path of dharma they would eventually arrive at Atman and Brahman anyway. This is also why Hindus consider Buddha to be a grand deceiver, those who were wordly that did not follow dharma in his time, could only be taught by coming down to their level.
