I don't know much about Hinduism except for some books I have read about Himalaya.
Buddhism rejects the idea of "soul" which makes it very different from Hinduism.
I never denied what ancient India achieved but I do not think Buddhism can fall under the category of "religion". Also just because Buddhism reached East Asia from India it does not mean that the version of Buddhism in India and versions of Buddhism in East Asia are the same. Buddhism changes its form as it reaches from country to another. It adopts into foreign cultures easily.
Buddhism is a religion and a way of life, this is a Dharmic religion and is not based on any books.
There will always be variations in any religion and some practices through out the world and Buddhism is no exception.
Similarities:
Hinduism and Buddhism share some of the following similarities.
1) Both Hinduism and Buddhism emphasize the illusory nature of the world and the role of karma in keeping men bound to this world and the cycle of births and deaths.
2) According to the Buddha, desire is the root cause of suffering and removal of desire results in the cessation of suffering. Some of the Hindu texts such as the Upanishads (Isa) and the Bhagavadgita consider doing actions prompted by desire an attachment would lead to bondage and suffering and that performing actions without desiring the fruit of action would result in liberation.
3) Both religions believe in the concept of karma, transmigration of souls and the cycle of births and deaths for each soul.
4) Both emphasize compassion and non violence towards all living beings.
5) Both believe in the existence of several hells and heavens or higher and lower worlds.
6) Both believe in the existence of gods or deities on different planes.
7) Both believe in certain spiritual practices like meditation, concentration, cultivation of certain bhavas or states of mind.
8) Both believe in detachment, renunciation of worldly life as a precondition to enter to spiritual life. Both consider desire as the chief cause of suffering.
9) The Advaita philosophy of Hinduism is closer to Buddhism in many respects.
10) Buddhism and Hinduism have their own versions of Tantra.
11) Both originated and evolved on the Indian soil. The founder of Buddhism was a Hindu who became the Buddha. Buddhism is the greatest gift of India to mankind.
B) Differences:
Following are some of the differences in the principles and practice of these two religions.
1) Hinduism is not founded by any particular prophet. Buddhism was founded by the Buddha.
2) Hinduism believes in the efficacy and supremacy of the Vedas. The Buddhist do not believe in the Vedas or for that matter any Hindu scripture.
3) Buddhism does not believe in the existence of souls as well in the first cause, whom we generally call God. Hinduism believe in the existence of Atman , that is the individual soul and Brahman, the Supreme Creator.
4) Hinduism accepts the Buddha as an incarnation of Mahavishnu, one of the gods of Hindu trinity. The Buddhist do not accept any Hindu god either as equivalent or superior to the Buddha.
5) The original Buddhism as taught by the Buddha is known as Theravada Buddhism or Hinayana Buddhism. Followers of this do not worship images of the Buddha nor believe in the Bodhisattvas. The Mahayana sect considers the Buddha as the Supreme Soul or the Highest Being, akin to the Brahman of Hinduism and worship him in the form of images and icons.
6) The Buddhists consider the world to be full of sorrow and regard ending the sorrow as the chief aim of human life. The Hindus consider that there are four chief aims (arthas) in life which every being should pursue. They are dharma (religious duty), artha (wealth or material possessions), kama (desires and passions) and moksha (salvation.)
7) Hindus also believe in the four ashramas or stages in life. This is not followed in Buddhism. People can join the Order any time depending upon their spiritual preparedness.
8) Buddhists organize themselves into a monastic Order (Sangha) and the monks live in groups. Hinduism is basically a religion of the individual.
9) Buddhism believes in the concept of Bodhisattvas. Hinduism does not believe in it.
10) Buddhism acknowledge the existence of some gods and goddesses of Hindu pantheon, but give them a rather subordinate status.
11) Refuge in the Buddha, the Sangha and Dhamma are the three cardinal requirements on the eightfold path. Hinduism offers many choices to its followers on the path of self-realization.
12) Although both religions believe in karma and rebirth, they differ in the manner in which they operate and impact the existence of individual beings.
you do provide a different version. But of course, with each reply, everyone giving a separate source of his origin. I shall also provide one. 
