Basically hindi is an evolved version of sanskrit and urdu has evolved from persian and arabic. In a laymans language, hindi is more or less a language developed for communication whereas urdu developed as poetic and artistic language. In india for a long time hindi was considered a language used for lower class like workers and peasants whereas urdu was considered a language used by upper class like nawabs and novels, just like english and latin in western countries. But like english hindi evolved and urdu has disappeared like latin though speaking and writing urdu is much more romantically involving just like french. Also as english has added latin words, hindi has added urdu in its language usage and its very hard to differentiate between hindi and urdu words.

I would like to add to this discussion that Hindi and Urdu also differ in the script see this example, I have written Hindi in devanagari script = हिंदी and this word is written in Urdu ہندی Both words are same "Hindi" but script is different. Hindi is written just like English from left to right while Urdu is written from right to left. About the similarities in two, much has already been said in the above comments. Arvind said that Hindi was considered a language used for lower class but its not completely correct as in the time of Aryans and kings after them Hindi was the main language, (watch or read Ramayan and Mahabharat) Hindi and Sanskrit have been around for a long time. Urdu came with the Muslim & Persian rulers. Another point is that as Hindi evolved it added words not only from Urdu and Arabic but from many different languages. Hindi is a language which keeps growing and including words from different languages around the world. Hindi is the national language of India but more prevalent in North India only; in Southern India, people speak regional languages like Tamil, Kannada but some also understand Hindi as many North Indians are also living there. I will end my note now but keep in mind that Hindi may seem difficult when you start but as you keep learning it becomes easy as its a phonetic and scientific language.

They differ vastly in literary and formal vocabulary; where literary Urdu draws heavily on Persian and Arabic, literary Hindi draws heavily on Sanskrit and to a lesser extent Prakrit. The grammar and base vocabulary (most pronouns, verbs, adpositions, etc.) of both Urdu and Hindi, however, are the same and derive from a Prakritic base, and both have a heavy Persian influence