Is there a connection between Lord Buddha & Hinduism?

Just would like to highlight the following things happened with respect to the ongoing discussions. Quotation from Wiki:

In 1999, at the Maha Bodhi Society in Sarnath, Jagadguru Sankaracharya, Jayendra Saraswati of Kanchi matha and Vipassana Acharya S. N. Goenka after having a mutual discussion, gave a joint communiqué agreeing on the following three points.

1. Due to whatever reason some literature was written in India in the past in which the Buddha was declared to be a re-incarnation of Vishnu and other various false things about him, this was very unpleasant. In order to foster friendlier ties between Hindus and Buddhists we decide that whatever has happened in the past should be forgotten and such belief should not be propagated.

2. A misconception has spread in the neighbouring countries that the Hindu society of India is organising such conferences to prove its dominance over the followers of the Buddha. To forever remove this misconception we declare that both Vedic and Samana are ancient traditions of India (Vishnu belongs to the vedic tradition and Buddha belongs to the Samana tradition). Any attempt by one tradition to show it higher than the other will only generate hatred and ill will between the two. Hence such a thing should not be done in future and both traditions should be accorded equal respect and esteem.

3. Any body can attain high position in the society by doing good deeds. One becomes a low person in society if one does evil deeds. Hence anybody by doing good deeds and removing the defilement’s such as passion, anger, arrogance, ignorance, greed, jealousy and ego can attain a high position in society and enjoy peace and happiness.

Courtesy quoted in an article in a blog https://karutharangam.blogspot.jp/2013/06/

There is no connection between Buddha and hinduism. Buddha the awaken and intelligent had denied Vedas and the brahmanas which were the main religious books of Hindus saying that it was not based on facts and logics. He said no to veda then how can he tell aomething about bhagavatgeetha.. bhagavatgeetha is simply a story what did he had to do with it. He said no to chaturvarna and spoke about equality but all Hindu mythology is about varna system and suppress and oppress women and shudras. Buddha was doing a social reform giving equality and you talk about the Hindu Vishnu incarnation which is totally irrelevant. He says there are no janmas and karmas of past life. There is only one life and everyone should go in the righteous path. He totally deneid the idea that god created us or there was a creator. He was an atheist. His teaching never said anyone to worship god instead to leave all the dogmas that Hindu had already put in the minds of people and said to evolve and be intelligent. Buddha means awaken. Anyone who lives in the natures principle and practising it can be buddha. He never mentioned buddhism anywhere.

When Gotama was alive, Hinduism did not even exist. The religion of the Brahmans at that time is best called Brahmanism. The Pali suttas show the Buddha knew about the Four Vedas of Brahmanism. Both an examination of the Vedas plus an examination of the interactions between the Buddha & Brahmans shows there was no systematic doctrine of karma & reincarnation in the Vedas. It appears the Brahmans believed they were born a special group from Brahma's breathe or mouth and would return to Brahma in heaven.

Therefore, what became known as Hinduism, with its various doctrines, such as; (i) the desireless Bhagavada Gita; (ii) the Trimurti of creation (Brahma), maintenance (Vishnu) & destruction (Shiva) and (iii) reincarnation over many lifetimes until reuniting with Brahma (whatever it is; was probably influenced by Buddhism & the later self-corruptions of Buddhism.

I think the common view that the Buddha was influenced by Hinduism is completely the opposite of the truth; that the truth is Hinduism was influenced by Buddhism.

Prior to the Buddha (born 563 or 480 BCE), was the historical Vedic religion (1750 - 500 BCE), followed by the shramanic movement (500 - 200 BCE), and the beginnings of the Upanishadic movement within Hinduism (500 - 200 BCE). Buddhism is also considered to be part of the shramanic movement. Vedanta came later as part of Classical Hinduism (200 BCE - 1100 CE). You can find this information here.

Prince Gautama (therefore a kshatriya, not brahmin), after leaving home, followed the shramanic traditions which were based on asceticism. Jainism was one of the major shramanic traditions at the time. The Pali Canon (DN2) has descriptions of various shramanic teachings at the time (Pūraṇa Kassapa, Makkhali Gosāla, Ajita Kesakambalī, Pakudha Kaccāyana, Sanjaya Belatthiputta and Mahavira). These teachings may or may not be compatible with Hinduism at the time.

Gautama tried to follow the different teachings at the time (Hindu and shramanic teachings) and found that it did not lead to the goal that he sought. And you know the rest of the story of how he found enlightenment and started teaching.

The Buddha did not accept the authority of the Vedas, preached that a Supreme Creator God did not exist and also taught anatta, which implied that there is no permanent absolute self (atman). All these contradicted the teachings of Hinduism at his time. The teaching of anatta and the Buddhist version of karma also contradicted Jain teachings.

According to the Krsna Consciousness (ISKCON) movement on this page, the Buddha, the Avatar of Vishnu, gave false atheistic teachings to the hedonistic nihilistic atheists who were living a sinful lifestyle, so that they can practise virtues, while remaining atheists, because God was kind to them and knew that they would never accept God and the "true" teachings which has both virtues and devotion to God. He also preached against the sacrifice of animals which was an abuse of Vedic sacrificial rites. The Krsna devotees therefore respect the Buddha but reject his teachings.

The idea that the Buddha is an Avatar of Vishnu comes from the Hindu text Bhagavata Purana, which was written around 500 - 1000 CE. It is possible that this was one way to bring Buddhist converts back to Hinduism, while reconciling the Buddha's position.

Did the Buddha say that he was an Avatar of Vishnu? No. According to the Buddhavamsa, he was the Bodhisattva Śvetaketu who resided in the Tushita heavans, before being born as the human Prince Gautama. Please see this answer.

The Hindu text Brahma Sutras, written around 200 BCE, has some criticism of Buddhism as found in this question.

Later on, Buddhist teachings was found to influence Hindu philosophy, to give rise to Advaita Vedanta. Even the Indian Mahayana schools of Madhyamaka and Yogacara predate the Advaita Vedanta school and Adi Shankara by centuries. Please see this answer for details. This meeting point of Advaita and Mahayana, now has the highest level of philosophical compatibility between the two religions. In fact, Advaita philosophers were accused of being Buddhists in disguise. However, anatta still keeps the two religions apart, philosophically.

The Hindu philosopher and scholar Adi Shankara who expanded Advaita, was also responsible in reviving Hinduism once again in India, and may have caused the decline of Buddhism and Jainism in India. However, Buddhism spread outside India.