An article on Language Log on Hindi-Urdu :

Hindi and Urdu are variants of the same language characterized by extreme digraphia: Hindi is written in the Devanagari script from left to right, Urdu in a script derived from a Persian modification of Arabic script written from right to left. High variants of Hindi look to Sanskrit for inspiration and linguistic enrichment, high variants of Urdu to Persian and Arabic. Hindi and Urdu diverge from each other cumulatively, mostly in vocabulary, as one moves from the bazaar to the higher realms, and in their highest -- and therefore most artificial -- forms the two languages are mutually incomprehensible. The battle between Hindi and Urdu, the graphemic conflict in particular, was a major flash point of Hindu/Muslim animosity before the partition of British India into India and Pakistan in 1947.

As the article notes, there are multiple variants of Hindi and Urdu. Having learnt Hindi, I would mention that Hindi and Urdu as languages spoken as part of every day can be mutually comprehensible if the speakers choose their words carefully. There are, however, differences in script as well, and these differences make seamless communication harder. (For the problem of script, there is transliteration software available to transliterate English to Urdu and to Hindi.).

There is a prodigal visual difference between the Devanagari script (also called Nagari) used to write Hindi and the Perso-Arabic script ordinarily used to write Urdu. The Devanagari script of Hindi is"squarish,'' "chunky,'' "has edges'' -- conventional characterizations all -- written left to right, with words set off from each other by an overhead horizontal line connected to the graphemes and running from the beginning of the word to its end. The Perso-Arabic script of Urdu is "graceful,'' "flowing,'' "has curves,'' written right to left, with word boundaries marked as much by final forms of consonants as by spaces. The immediate visually iconic associations are: Hindi script = India, South Asia, Hinduism; Urdu script = Middle East, Islam. The graphemic difference between Hindi and Urdu is far more dramatic, for example, than the difference between the Cyrillic script of Serbian and the Roman script of Croatian.

Hindi and Urdu get more highly differentiated as we step up the High scale with each borrowing from distinct vocabularies. So High Hindi is quite different from High Urdu.

Common words like chai 'tea', milna 'to meet', and mashin 'machine' are the same in either Hindi or Urdu. Vocabulary diverges sharply as we move from Low to High. The Hindi words for 'south' and 'temperature' (as in weather) are dakshin and tapman, the Urdu words junub and darja-e-hararat. The sentence "Who is the prime minister at the moment?'' is ajkal pradhan mantri kaun hai? in Hindi, ajkal vazir-e azam kaun hai? in Urdu.

An Indian linguist has illustrated how far the styles deviate from each other by asking how the abstract expression "salvation's true path'' might be translated into Hindi and Urdu at different style levels and among different ethnic-social groups. Village people would render this as mukti-ki sacci sarak (Bazaar Hindustani). Pandits or educated Hindus would say mukti-ki satya upay (Highbrow Hindi). Cultured Muslims would translate the phrase as nájat-ki haqq rah (Highbrow Urdu). Indians who speak English as their second language might say salweshan-ki tru path. The only indication that these four "languages'' are in some sense variants of the same language is the genitive marker -ki. Words like satya and upay in the Highbrow Hindi rendering are from Sanskrit. Every single content morpheme in the Highbrow Urdu version is from Persian or Arabic. One sees how dramatically the character of a language is changed when the sources of borrowed words for new concepts are as far apart as they are in Hindi and Urdu: we might as well be dealing with different languages.

A minor point on diction would be in order here. Based on the Hindi that I have learnt, using the word "sarak" in that sentence would be inappropriate diction. "raah" (or maybe 'raastaa') would be my word of choice there. Quibbling aside, the article does make some very interesting comments : one, that Hindi and Urdu are two languages with digraphic differences; two, that digraphia can be a separative force; three, that there are high and low variants of Hindi-Urdu and that these language "diverge from each other cumulatively as one moves from the bazaar to the higher realms"; four, that this digraphia may have been perceived as a somewhat divisive issue between the different religions in the Indian subcontinent; and five, that there were political differences of opinion regarding script as well.

Gandhi's comments on the topic have much good sense.

Gandhi's tendency overall was to minimize the role of script. In a 1918 speech he laid out his thinking:

Hind[ustani] is that language which is spoken in the north by both Hindus and Muslims and which is written either in the Nagari or the Persian script. [It] is neither too Sanskritized nor too Persianized .... The distinction made between Hindus and Muslims is unreal. The same unreality is found in the distinction between Hindi and Urdu ... . There is no doubt or difficulty in regard to script. As things are, Muslims will patronize the Arabic script while Hindus will mostly use the Nagari script.