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The following is an excerpt from the Introduction to Afroz Taj’s book: Urdu Through Hindi: Nastaliq With the Help of Devanagari (New Delhi: Rangmahal Press, 1997)

South Asia is an area of enormous linguistic diversity. As one travels from town to town, or from region to region, accents change, dialects change, and languages change. The dividing lines between languages are often not clear, being blurred by a proliferation of overlapping and interlacing dialects. Even on the same street one can find very different “languages”: the engineering student will speak differently from the poet, the servant will speak differently from the priest.
What then defines a language? What makes a language unique? Its writing system? Its vocabulary? Or its grammatical structure?
We should not make the mistake of confusing a language with its writing system. Often, unrelated languages share a common writing system, while any language can be transcribed into a new writing system without affecting its basic sounds and structure. Indeed most of the world’s languages borrowed their writing systems from somebody else. For example, English, French, and Spanish borrowed their writing systems from Latin. The Latin alphabet was in turn derived from the Greek alphabet. Japanese adopted the Chinese word-characters. Both Indonesian and Turkish switched from the Arabic to the Roman writing system early in the Twentieth Century without being otherwise changed significantly. English can be written in Morse Code, Braille, binary computer code, or even in the Hindi writing system and it still remains English.
Vocabulary is likewise not characteristic of a language. Words are readily borrowed between languages like dry leaves blown about in the wind. No language can claim a pure, fixed and unchanging vocabulary. Indeed, it is often impossible to express oneself in English without using words borrowed from French, Latin, or even Hindi.
Thus it is only grammatical structure that can be said to characterize a language. No matter which writing system is used, no matter which vocabulary words are used, a language’s grammar will follow regular and characteristic rules. These rules, which govern verb conjugation, noun declension, plural formation, syntax, etc., are largely consistent within a language but differ between languages. Thus a comparative study of these rules allows us to distinguish one language from another.
Therefore Hindi and Urdu, which share a common, identical grammatical structure, must be considered a single language: Hindi-Urdu.
How did Hindi-Urdu develop, and why does it have two names? Let’s look at the linguistic condition of India about one thousand years ago. The Indo-Aryan language family, brought to South Asia by the Aryans in prehistoric times, had become firmly established in a belt running from the Persian Caucasus in the West to the Bay of Bengal in the East. The descendent languages of Sanskrit, including several dialects of early Hindi, Medieval Panjabi, Gujarati, Marathi, and Bengali, as well as their cousin tongue Persian were emerging in their respective regions. The Indian languages had adopted the ancient Sanskrit writing system, Devanágari, in various forms, while Persian had borrowed the Arabic writing system from its neighbors to the West. Hindi, in various dialects including Khariboli, Braj Bhasha, Bhojpuri and Awadhi, was spoken throughout North Central India.
Then, about seven centuries ago, the dialects of Hindi spoken in the region of Delhi began to undergo a linguistic change. In the villages, these dialects continued to be spoken much as they had been for centuries. But around Delhi and other urban areas, under the influence of the Persian-speaking Sultans and their military administration, a new dialect began to emerge which would be called Urdu. While Urdu retained the fundamental grammar and basic vocabulary of its Hindi parent dialects, it adopted the Persian writing system, “Nastaliq” and many additional Persian vocabulary words. Indeed, the great poet Amir Khusro (1253-1325) contributed to the early development of Urdu by writing poems with alternating lines of Persian and Hindi dialect written in Persian script.
What began humbly as a hodge-podge language spoken by the Indian recruits in the camps of the Sultan’s army, by the Eighteenth Century had developed into a sophisticated, poetic language.
It is important to note that over the centuries, Urdu continued to develop side by side with the original Hindi dialects, and many poets have written comfortably in both. Thus the distinction between Hindi and Urdu was chiefly a question of style. A poet could draw upon Urdu’s lexical richness to create an aura of elegant sophistication, or could use the simple rustic vocabulary of dialect Hindi to evoke the folk life of the village. Somewhere in the middle lay the day to day language spoken by the great majority of people. This day to day language was often referred to by the all-encompassing term “Hindostani.”
Because day to day Hindostani was essentially a widespread Indian lingua franca not associated with any particular region or class, it was chosen as the basis for modern Hindi, the national language of India. Modern Hindi is essentially Hindostani with a lexicon of Sanskrit-derived vocabulary in preference to the Persian borrowings of literary Urdu. Likewise, Hindostani in its Urdu form was adopted by Pakistan as a national language because Urdu is not tied to any of the regions comprising modern Pakistan. Thus a language which is really nobody’s mother tongue is today the second most spoken language in the world, understood throughout most of the populous Indian subcontinent and in many unexpected corners of the globe.
If we were to draw a picture of the relationship between all of these various versions of Hindi-Urdu, it might look something like this:

[Persian] Literary Urdu ——— Modern Hindi [Sanskrit]
Hindi-Urdu (Hindostani)
Hindi Dialects (Khariboli, Braj, Bhojpuri, etc.)

As for “Hindustani” that’s an archaic term - nobody identifies with that anymore.