The subcontinent of India was traditionally divided into many smaller states. In the north, the Muslims prevailed, ruling in the Sultanate of Delhi, but also in Bengal and the Deccan which split off from Delhi. In the south, Hindu states survived; the Chola Empire soon was replaced by the Pandyan Empire, which later itself was replaced by Vijayanagar, founded as a counterweight against the Muslims.
Generally, India developed with much less external influence; the Europeans arrived centuries later than in OTL, and there never were any invasions of Mongols or their successors. China, during the periods when it was stronger, exerted a certain influence in the NE however, and Persia actually managed to conquer Delhi later.
Science in India
The School of Kerala is responsible for many progresses in science, especially in mathematics, which were developed there earlier than anywhere else on Earth.
Religion in India
In India, the monotheist religions (various forms of Christianity and Islam) clashed with the eastern religions (Hinduism, Buddhism and even Confucianism). This lead to the development of various synkretist religions, like Sikhism and the teachings of Chandramoorthy, or even the cult of the emperors of New Rome.