Hinduism does not much disagree with Buddhism or its teachings which are by and large similar to ours. Hinduism and Buddhism are extensively cross-pollinated systems having arisen from the same fertile intellectual matrix.

The major differences centre around the abstract Self vs Non-self debate which is so arcane as to be irrelevant to daily life as lived by 99.9% of humans struggling with dukha (dis-ease, suffering, default low-level neurosis). The problem of suffering (dukha) being the pivotal focus of ALL schools of Indian Philosophy.

The proposition by both camps is that whatever we consider to be Self or a buttress to notions of self - i.e. name, form, sex, gender, race, class, nationality, qualification, relationships, interests, ideas, beliefs, thoughts etc. are all FALSE cognitive constructs and are of the nature of Spiritual nescience (avidyā) and the cause of all suffering.This much we all whole-heartedly agree in unanimity.

The next question is where the putative disagreement lies - once all of the ego-generated self-clinging (asmitā/ahaṅkāra) is stripped away what remains? Is there a thing or is there nothing? In Vedanta it is known as the iti-neti paradox (affirmation vs denial.)There are many different views and conjecture on the nature of the Ultimate Reality in the many schools of Hinduism and Buddhism but these arguments and debates are practically futile because both agree that the Ultimate Reality is beyond the ability of the mind to comprehend. Hence the Buddha avoided discussion on them and discouraged metaphysical speculation.

Buddha, wasn’t really interested in being an orthodox teacher or preacher in the Vedic tradition. But he wasn’t unique in that aspect. His time saw many such teachers and preachers (including Mahavira) who absorbed the Vedic doctrines floating around in the air while adding or removing elements or innovating new ideas and theories. Buddha himself explored many such theories while studying under many such teachers. He does reference ancient Vedic rishis as revered role models to aspire to. But beyond that, he doesn’t really feel the need to stay “faithful” to the Vedic tradition. So much that later Buddhism is clearly listed under “heterodox” schools of thought, as opposed to the “orthodox” schools of thought which revere Vedic tradition.

One must remember that both philosophical systems existed and flourished and developed in a common social and intellectual milieu. They studied each others’ texts, debated together, argued together and exchanged ideas. Throughout South-East Asia during the great Khmer empire both religions flourished side by side — the metaphysical differences were a matter for scholars, the common people had no idea that they were different religions.

Secondly the differences are actually very esoteric. For example the biggest contention is about Īśvara and Ātman.

Īśvara is the personal (saguna) aspect of Brahman — the undifferentiated (nirguna) Ground of Being. The Buddhists deny any personal deity. Contrary to popular opinion the Buddha never denied the existence of the Hindu gods, he mentions them, and after his enlightenment he was reluctant to preach until Brahma came and persuaded him to do so. In Thailand the monks preach the Dharma from behind screens with the face of Brahma on them. And Brahma is the most common image around!

What the Buddha preached was that the gods had no hand in the Liberation (nirvāṇa) of humankind and we all have to strive for it ourselves, relying only on our personal effort.

The issue of the ātman (Self) is complex. Hindus claim that we are all modes of consciousness (jīvātmans) that have taken birth in human bodies due to Karma. There are three Hindu schools of thought on this matter;

1. The Advaitians (non-dualists) who claim that all the ātmans are one in essence and ultimately unite and coalesce in Brahman in the state of mokṣa/nirvāṇa.
2. The Visiṣṭhādvaitins (qualified non-dualists) who claim that the ātmans are one in essence but retain some degree of atomic separation in mokṣa/nirvāṇa.
3. The Dvaitins (dualists) who claim complete and eternal separation of ātmans, always.

The Buddhists hold with the ‘non-self’ doctrine (anātman). The Buddha said: “I have searched everywhere (body and mind) and I have failed to discover a Self!” The Hindu polemicists would rejoin:– “well then! Who was it that was doing the searching???”

The anātma doctrine denies only the substantial existence of the ātman as a unique entity not the existential experience of a Self. So in this wise it is in agreement with Advaita Vedānta (often called by detractors as pracchana bauddha-vāda — crypto-buddhism).Advaita Vedanta has many similarities with Madhyamika Buddhist school of thought.

Other differences revolve around the nature of consciousness, what is it that actually incarnates? If there is no Self then how does karma transfer from one life to the next? But as I said, these metaphysical issues so loved by the scholars are irrelevant to the devotees in the temples, ashrams and monasteries.

The biggest difference to the laypeople of the Buddhist region is Buddhism is a monastic based religion and has very little to offer to householders except the precepts. There are no Buddhist rituals for weddings, funerals, birth, passages of life etc. So wherever Buddhism has gone these lay rituals were either carried over and modified from Hinduism or adopted from the indigenous people - like Shintoism in Japan.

Buddhists and Hindus are not rivals and are not competing for converts or privilege.We happily visit each others temples respect each others clergy and listen to each others teachings which are not all that different. For most part there is a symbiosis.

The Buddhist king of Thailand is still regarded as an incarnation of Vishnu, and entitled RAMA. The Ramayana is depicted on the walls of the royal palace. All the Buddhist countries have their own version of the Ramayana. The Royal temple is Hindu with statues of the trinity: Brahma, Vishnu and Shiva. All the ceremonies and protocols as well as the recent cremation of the king are done by the royal Thai Brahmins according to Hindu rituals.

The Tibetans adopted Vedic rituals like the Yajña (fire oblations) and perform them with great panache! Most of their ceremonies are Hinduesque . Even the Shinto religion has many of the Vedic Hindu deities and the fire-ceremony (called GOMA) is popular and beautifully executed (more aesthetic one may say than in India!).Thailand which is one of the premier Buddhist countries is the best example of Hindu-Buddhist harmony - the King is a Buddhist and the Royal preceptor (rāja-guru) is a Brahmin. The palace in Bangkok contains a Hindu Devasthānam (temple) dedicated to the Hindu Trinity and the (Thai) Brahmins chant Tamil hymns of the Tirupavai.

The vast majority of Hindus don’t have critical opinions of the followers of other religions but just accept them and live with them as they are - just like most humans do.

For the philosophers, priest, monks and theologians there are some slight differences but far more similarities - so we are on very amiable and close relations - I dare say there is far more camaraderie between Hindu and Buddhist clergy than there is between the monotheistic clergy.

Historically Hindus and Buddhist have intermingled and blended throughout South East Asia. Even in Srilanka in spite of the recent ethnic strife, Buddhists and Hindus have lived together in peace and cooperation. An example of the inter-faith harmony is the Kataragama temple which is a common sanctuary. I myself have performed many inter-faith marriages between Hindus and Buddhist and have never encountered any form of disagreeableness or tension.

Politically there is now a tendency among some “Neo-buddhists” - Navayanas - who are mostly ex-Hindu (Dalits), to weaponise religion and to try to drive a wedge between the two communities.

The major differences lies in ATMAN (SELF) vs ANATMAN(NON-SELF)
There are hundreds of passages in the Pali Tripitaka which have parallels in the Upaniṣads and related texts, so the Buddhas teaching was not new but it was unique.

The controversy of the ātma (self) vs anātma (non-self) is simply vexatious and in my experience is stressed more by western Buddhists than traditionalists.

In Sanskrit ātma has several meaning as illustrated by this verse in the Gītā

uddhared-ātman-ātmānaṁ nātmānam avasādayet |

ātmaiva hy-ātmano bandhur-ātmaiva ripur-ātmanaḥ ||

Translation:– One should raise one’s Self by one’s own mind and not allow one’s Self to sink; for the mind alone is the friend of the Self, and the mind alone is the adversary of the Self.

So ātma means self as in “I myself did it” or it can mean “mind” or it can mean “Self” as an individual mode or unit of consciousness. It can also refer to the Absolute Brahman or the Supreme Being (paramātman)

And then there are 2 levels of TRUTH promulgated both by Buddha and the Rishis - vyavahārika satya (conventionally or everyday truth) and paramārthika satya (Absolute Truth.)

So in Advaita Vedānta and in Patanjali Yoga there is a “self” in the everyday conventional sense but “non-self” in the absolute, abstract sense in which individuality is transcended. There are many Vedānta texts which also talk about No-mind.

So one needs to examine the sense and context in which the Buddha was teaching.

In the Annatta-lakhana sutta mahavagga 1:6:38 the Buddha says:–

'Therefore, O Bhikkhus, whatever body has been, will be, and is now, belonging or not belonging to sentient beings, gross or subtle, inferior or superior, distant or near, all that body is not mine, is not me, is not my Self: thus it should be considered by right knowledge according to the truth.”

So what the Buddha is saying is that what you think of, or identify as “self” - body/mind, complex is not YOU. This is exactly the same as what the Rishis of the Upanishads and the Bhagavad Gita are saying. When the Buddhist says - “I looked everywhere but could not find a self (ātma)” The Vedāntin asks “who is doing the looking?” There cannot be a process without an observer.

Now we have determined that the mind/body is not the true Self in both Vedanta and Buddhism and that somehow our true identity is related to our sentience or consciousness and our perception, the question arises - so what is that “observer” or sākṣi. Different opinions are held by different teachers.

Some say that the sākṣi is simply a delusion of separation and that with the dawning of enlightenment (bodhi) all individual consciousness coalesce into a homogeneity of unified consciousness.

Others say that the units or modes of consciousness merge like atoms of h2o merging into the sea.

Others say that the units of consciousness remain eternally separate like grains of rice in a pile.

The Buddha never encouraged metaphysical speculation since it is beyond the capacity of humans to conceive or to articulate - a sentiment echoed in the Taittiriya Upanishad.

The major difference I find lies in this - the Buddha’s refusal to speculate and Tthe Hindu sages attitude - “well yes it is inconceivable and inexpressible but we’re going to speculate and articulate it anyway!!

So, if the Buddha indeed denied the existence of a Self then who, or what is it that experiences Nirvāṇa?

The Buddha gives 30 metaphors for nirvana/nibbana, an otherwise inconceivable (non)state: the taintless, the truth, the far shore, the subtle, the very difficult to see, the un-aging, the stable, the un-disintegrating, the unmanifest, un-proliferated, peaceful, deathless, sublime, auspicious, secure, the destruction of craving, wonderful, amazing, the unailing state, nonbinding, unafficted, dispassion, purity, freedom, un-adhesive, the island, the shelter, the asylum and the refuge. (SN 43)

None of these refer to nihilism or a doctrine of extinction - so the question is who is the subject of these adjectives?

Buddha said “i have searched everywhere but I have not found a self” - so who was doing the looking-Hindus would like to ask? He also taught a positive state of Nirvana - so who or what experiences Nirvana?Many Buddhist monks themselves clarify the anatma theory and say it has been misunderstood.

For Understanding this ANATMA doctrine ,Let us take

Annattalakhana Sutta Mahavagga 1, 6, 38 -  And the Blessed One thus spoke to the five Bhikkhus: 'The Body (Rupa), O Bhikkhus, is not the Self. If the body, O Bhikkhus, were the Self, the body would not be subject to disease, and we should be able to say: " Let my body be such and such a one, let my body not be such and such a one." But since the body, ‑ O Bhikkhus, is not the Self, therefore the body is subject to disease, and we are not able to say: "

Let my body be such and such a one, let my body not be such and such a one." 'Sensation (Vedana), O Bhikkhus, is not the Self,.... Perception (Sanna) is not the Self, . . . The Mental formations (Sankharas) are not the Self .... Consciousness (Vinnanam) is not the Self.

Nirvana is not just a negative state of cessation of suffering it is a positive state as well - "This is peace, this is exquisite -- the resolution of all fabrications, the relinquishment of all acquisitions, the ending of craving; dispassion; cessation; Nibbana." -- AN III.32

"There is that dimension where there is neither earth, nor water, nor fire, nor wind; neither dimension of the infinitude of space, nor dimension of the infinitude of consciousness, nor dimension of nothingness, nor dimension of neither perception nor non-perception; neither this world, nor the next world, nor sun, nor moon. And there, I say, there is neither coming, nor going, nor stasis; neither passing away nor arising: without stance, without foundation, without support [mental object]. This, just this, is the end of stress." - Ud VIII.1

Hindu Doctrine of ATMAN -

If we compare it with Hindu Upanishads.Here is a description of ATMAN from Hindu Upanishads- 

Not inwardly cognitive, not outwardly cognitive, not both-wise cognitive, Not an undifferentiated mass of consciousness, not cognitive, not non-cognitive, Unseen, with which there can be no dealing, ungraspable, devoid of characteristics: Inconceivable, indefinable, its sole essence being the consciousness of its own Self, The cessation of proliferation, tranquil, blissful, without a second; [such] they consider is the fourth [state of being]. This is the Atman [Self]. This should be realised.”

Mandukya Upanishad 7-

So the Hindus would like to ask Buddhists

.Who experiences the peace and joy and absence of craving which is Nirvana?Tell me what it is that experiences cessation of suffering and the peace of Nirvana - gotcha! All investigative ideologies including Science come to a point of “unknown” in relation to first things. Why the Bigbang? Why is there something rather than nothing? - No one can answer these question not you nor I nor any other religion or science. 

Hinduism says Sarvam Khalvidam Brahman which Everything is Brahman and It is the material cause of Universe

These are the 4 Main Mahavakyas in Vedanta

1 - Sarvam khalvidam Brahman — Everything that one can sense with the five senses in Brahman (meaning the “immensity” _ the “Unified Field”, the “ground of Being”.)

2- Prajñānam Brahma — that Immensity or Unified Field is consciousness.

3. Ayam ātma Brahman — this individual Self (ātman) which is me in my essential nature as a ray of consciousness is that same Unified Field — Brahman.

4 - Tat tvam asi — “you are that” – you, me and ever sentient being is a ray of that Supreme All-pervading consciousness.

On Close observation-the differnces seem to be very futile n they both sound as same