Advaita is one school of Vedanta and not the whole of it, though its prominence in Western accounts of Hinduism can obscure that. The other schools read the same texts and reach different conclusions.
Ramanuja, in the eleventh century, argued for vishishtadvaita, qualified non-dualism: there is one reality, but it is internally differentiated, and individual souls and the material world constitute the body of God, real and permanently distinct from him as a body is distinct from the self that animates it. His motive was devotional as much as logical, since a love relationship requires two parties, and he objected that Shankara's scheme makes worship a stage to be discarded.
Madhva, in the thirteenth century, argued for outright dualism, dvaita: God, souls and matter are eternally and really distinct, souls are eternally dependent, and the difference is never overcome.
The quarrel with Buddhism was fiercer than either. Advaita and Madhyamaka can look similar from outside, since both dissolve the world of ordinary distinctions, and Shankara's opponents accused him of being a crypto-Buddhist. He was at pains to deny it, and the denial identifies the real difference: Advaita affirms one absolutely real thing, being, consciousness and bliss, whereas Madhyamaka denies inherent existence to everything without exception and refuses to install any ground. Shankara therefore charged the Buddhists with nihilism; Buddhist logicians charged Advaita with reinstating the self that Buddhism was founded on rejecting. Neither charge is fair on the other's own terms, and both were meant.
Advaita reached the modern West largely through Vivekananda, who presented it at the 1893 World Parliament of Religions as the philosophical core of Hinduism, and through Ramana Maharshi and the neo-Vedanta teachers who followed. That presentation has shaped the widespread impression that Hinduism is monist by default, which the existence of Ramanuja, Madhva and the whole devotional mainstream contradicts.