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, 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 the Vedic religion at the time.
Gautama tried to follow the different teachings at the time (Vedic 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.
Hindu religious scholars today like to do two things. The first is to place all these different groups under the umbrella of Hinduism, sometimes including religions that do not accept the authority of the Vedas like Buddhism and Jainism, but sometimes not. This is often used to generalize that all concepts in Hinduism today are older than Buddhism, when in fact, not all concepts in today's Hinduism come from the ancient Vedic religion. On the other hand, Buddhist scholars tend to describe Classical Hinduism as "Hinduism", while they describe the Vedic religion as "Brahmanism".
By grouping everything together, the Hindu religious scholars hide the fact that Buddhism (and other shramanic schools) had indeed influenced Vedanta and Classical Hinduism. It is well known that the founders of Advaita Vedanta including Gaudapada and Adi Shankara, were influenced by Buddhist philosopher Nagarjuna's Mula-madhyamaka-karika and Buddhism. This can be seen in Hinduism.SE questions here and here.
The second thing that Hindu religious scholars like to do is misrepresent the age of a scripture. The Brahma Sutras or Vedanta Sutras which includes criticism of Buddhism (see this question) is claimed to be much older than the Buddha by Hindu religious scholars (see here). Meanwhile, in the Wikipedia article on this text, we can see that most historians and academic philosophers have the opinion that the Brahma Sutras or Vedanta Sutras come at least two centuries after the Buddha. The same applies to other Hindu texts like the Bhagavad Gita and certain Upanishads (Kena, Katha, Isa, Svetasvatara, Mundaka, Prasna, Mandukya), according to the timeline of Hindu texts.
This again shows that many writings that come after the Buddha, would most likely have experienced influence from the Buddha's teachings, but would be misrepresented by Hindu religious scholars as being far older than the Buddha.
While the ancient Vedic religion was based mainly on sacrifice and worship to the Vedic gods, I would guess that ideas like dhyana and samadhi come from the shramanic movement, which include ascetics who may or may not accept the authority of the Vedas. It is also likely that shramanic ascetics may have influenced each other, just as the Buddha learned some things from other teachers like Alara Kalama and Uddaka Ramaputta, although they did not achieve the Buddha's level of enlightenment before the Buddha did. It is well known that the Buddha discovered Vipassana meditation but not Samatha meditation (see this answer). Samatha meditation existed before the Buddha's time.
Apart from that, there are also other concepts that are simply common human themes that can be found in many cultures around the world, or are common words found in many languages. Examples include karuna (compassion), atma or atta (self), maitri or metta (friendliness).
Here are some major doctrinal differences between Buddhism and Classical Hinduism:
- The middle way between eternalism and annihilationism (Hinduism subscribes to eternalism of the self) - see this answer
- The self is not eternal, not standalone and not independent - see this answer, and SN44.10, and contrast with Hindu BG2.24
- The self or soul does not pervade the body (unlike the Hindu BG2.17) - see SN35.85
- There is no Supreme God (unlike Hinduism) - see this answer
- The self or soul does not transmigrate (unlike Hindu BG2.22) - see this answer
- Warriors dying on the battlefield while performing their duty do not go to heavan after death according to SN42.3 (unlike Hindu BG2.32)
- Lay people eating meat that was bought dead and frozen from the supermarket is not sinful - see this answer
Centuries after the Buddha's passing, the development of Buddhism and Hinduism did influence each other. As discussed in this answer, later on, Buddhist and Hindu philosophy influenced each other to produce Advaita Vedanta and Indian Mahayana Buddhist philosophy. Also, Tibetan Buddhism have adopted some Hindu deities. On the other hand, there has also been debates in later times between Hindu and Buddhist scholars (see this question).
It must be clear that the Buddha's unique contributions that cannot be found in Hinduism and other religions, are anatta, vipassana, dependent origination (pratityasamutpada) and the middle way between eternalism and annihilationism (see this answer).
Furthermore, anatta is the irreconciliable difference between Buddhism and Hinduism, as stated by eminent German indologist Professor Helmuth von Glasenapp in his essay "Vedanta and Buddhism: A Comparative Study":
Nothing shows better the great distance that separates the Vedanta and
the teachings of the Buddha, than the fact that the two principal
concepts of Upanishadic wisdom, Atman and Brahman, do not appear
anywhere in the Buddhist texts, with the clear and distinct meaning of
a "primordial ground of the world, core of existence, ens realissimum
(true substance)," or similarly.