It comes down to two linguistic concepts: Abstand and Ausbau. Abstand is how similar two varieties are (If I'm speaking Urdu and you're speaking Hindi, how well can we understand each other?); Ausbau is how close two communities that use different varieties are from each other (If I'm from a community that speaks Urdu and you're from a community that speaks Hindi, how much do our neighbours mind if a stranger mistakes them for one another?)

The answer to the first question is that we can understand each almost perfectly, so there's little Abstand distanfe, if any at all. The answer to the second question is that the history of South Asia over the last century would be a lot more peaceful if Hindi and Urdu speakers got along better, indicating quite a large Ausbau distance.

When studying varieties of languages (be they language or dialect), the symbols used to represent the sound are almost never studied, because that's a purely political decision. Urdu and Hindi use extremely different symbols to represent the sounds because their speakers are extremely intent on highlighting their differences (Ausbau). The actual sounds represented by those symbols are of much more importance when studying languages.

Technically, Hindi and Urdu are both varieties of Hindustani, but standardised along different registers. From the middle of the 16th Century, Urdu and Hindustani were used interchangeably to refer to the same language variety. Late in the 19th Century, Hindu (religion) speakers of Hindustani (language) wanted to distinguish themselves from Muslim speakers of Hindustani, and so created a standard that was more heavily influenced by Sanskrit and featured much less Persian influence, which they called ‘Hindi’.

Hindustani, or Hindu-Urdu, had historically been written in the Nastaliq script, which was impoted from Persia and the Middle East and was therefore associated with Islam, while Sanskrit used to be written in the Devnagri script when it had been in use and was associated with Hinduism. So when the new register for Hindustani was developed with all its Sanskrit influence, the script adopted by the Hindu community was the same as that used by Sanskrit.

Coming back to your question, the reason the two language varieties sound so similar is because they're both varieties of speech derived from the same parent variety. The reason they have such different scripts is because the communities that use these language varieties wanted to focus on how different their communities were from each other.