Both descend from ancient Vedic religions. They share an appreciation of images, as aids to worship, meditation, and tantra behaviors. Tantra behaviors were and are a form of behavior with rules of engagement intended to create a relationship with the Holy by sexual behavior between people; they are not excuses for sexual promiscuity as Western world- views would see the sexual behavior.
Both agree that one is reborn again and again, and that it would be good to escape this compulsory rebirth. Offen, and simplistically called 'reincarnation.' When one no longer is subject to the compulsory rebirth, one's Western soul is released to something better, but not at all like the Christian heaven.
I(t's release; not achievement of a better life, more fun, more things, or again simplistically, lots of sexual opportunity previously missed. It's release. )
Hinduism has emphasized that the Ultimate is Full (purna in Sanskrit) and rich in characteristics. Buddhism has emphasized how it is Empty (shunya) of anything that is permanent, causeless and separate.
The disagreement focuses in the idea of a self: Hinduism says we have one, and our essential self is the Self of the universe. Buddhism has said there is no abiding self, either for us or for the universe - all is flow. So Hinduism (for example) has a rich and loving temple and home ceremonial life around the manisfestation of the Self in many forms. Buddhism is more monastic. Buddhism loves meditation on the changing, vanishing, re-arising impermanent breath.
They had this argument in India, then Buddhism (dependent on its monasteries) died out during the Muslim invasion in India. So now another difference is that Hinduism is most common in India, and Buddhism is most common elsewhere.
____________________________________________________________________________________
Buddism started with a rebel prince called Gautama Siddartha around the time of approximately 500 BC. Gautama rebelled against the oppressive socio-religious (Hinduism) practices of his royal family. He rebelled against the brutal Vedic rituals that involved the slaughter of human and animal victims.
Gautama was terrified of the natural human ailments and sufferings such as diseases, old age and death. He abandoned his wife Yasodhara and his infant son Rahula to seek spiritual enlightenment. He had a number of teachers including the Jains. He challenged most of the teachings to which he was exposed, until he finally settled down and became a monk.
Guatama went on to become the founder of Buddhism and he was called The Buddha.