Buddha Purnima (or Vesak), which was designated as such by the Bihari Buddhist for Siddhartha Gautama, celebrates a number of events-conception, birth, enlightenment and death by which he became known as the Buddha. Buddhism, therefore, is worthy for itself understanding and even parallels in value to understanding what Buddha is to Hinduism.

The Hindu Roots of Buddha

Siddhartha Gautama was born a Hindu prince of the Shakya clan in Lumbini (Nepal today) around 563 BC. King Suddhodana, his father, who according to Hinduism tradition, invited the learned Brahmins to forecast the future of his son. The sage Asita declared, the prince would either become one of the great kings or a true spiritual leader if he witnessed suffering. 

The Bhagavata Purana (1.3.24) accepts Buddha as the ninth avatar (incarnation) of Lord Vishnu: "tataḥ kalau sampravṛtte sammohāya sura-dviṣām buddhonāmnāñjana-sutaḥ kīkaṭeṣu bhaviṣyati" Translation: "Then, in the beginning of Kali-yuga, for confusion of the enemies of the demigods, [the Lord] will appear as Buddha, the son of Añjanā, in the province of Gayā." 

Then the path of enlightenment

In a palace of indulgent luxury, Siddhartha drew no distinction between what was pleasure and what suffering in life until he was twenty-nine and ventured outside; then he saw the "four sights," which Hindu texts describe as an old man, a sick person, a corpse, and an ascetic. These significant encounters brought forth the basic sufferings dealt with in Hindu philosophy-old age, sickness, and death-and subsequently provoked him toward a spiritual quest. 

He followed the traditional Hindu ascetic practice at first. The Lalitavistara Sutra gives account of his training possibly by Alara Kalama and Uddaka Ramaputta to learn intensely concentrated meditative states. Failing to attain complete liberation from this, he then took upon himself severe penances (tapas) rigorous by those Hindu ascetics around at that time.

After six years of grievous self-mortification, he found that even this didn't liberate him. Middle of these two extremes-both indulgence and extreme asceticism-leads to enlightenment under the Bodhi tree at Bodh Gaya, nirvana, a term already found in Upanishadic literature.

Buddha's Wisdom 

Buddha incorporated or redefined much of Hindu doctrine in his own teachings, clean from Islam:

Dharma: if Hinduism considers dharma as sacred duty corresponding to one's station in life, then in contrast to this, Buddha emphasized dharma as universal truth and right conduct

Karma: Similar in both traditions but emphasized intentional acts in Buddha; mere ritualistic actions were unimportant to him.

Samsara:He seems to accept the Hindu conception of rebirth, but he does not think that atman (soul) is any way permanent. 

So both are about liberation from suffering but in a different way-Moksha/Nirvana. The Dhammapada collected Buddha's essential teachings and philosophically was close to the Upanishads: "Mind precedes all mental states. Mind is their chief; they are all mind-wrought." (Dhammapada 1). It reflects the Upanishadic teaching: "You are what your deep, driving desire is. As your desire is, so is your will. As your will is, so is your deed.