Contrary to popular belief, Sanskrit is not the original Indian language. As most educated Indians will know, the original inhabitants of India were the Dravidians. When the Aryans invaded, the Dravidians were driven down south, to the states known today as Tamilnadu, Kerala, Andhra Pradesh, Telangana, and Karnataka. The languages spoken in all these states stem from Tamil, which is a Classical Dravidian language, and therefore, Tamil stands as THE original Indian language. Modern Tamil is quite different from the original Tamil of course, but not too far off. By contrast, Sanskrit and its derivatives, Hindi included, were brought in by the Aryans. Of course, cultural assimilations over the ages led to the development of many words in Tamil from Sanskrit, and some words in the Sanskrit-based languages have some Tamil influence.
So to answer your question, the reason many Tamilians (such as myself) don't speak Hindi is simply because it is what distinguishes us from the Northern states. It is our culture, our identity, and we're proud of it. We aren't opposed to learning Hindi per se (I myself find it quite difficult to get by in the Northern states since I speak very little Hindi), but let me ask all northerners one thing: Why do you not speak Tamil? I agree, we may refer to Northerners as Hindiwalas, but then again, you call us all Madrasis, so......
We also need to get rid of the misconception that Hindi is the national language, because it isn't. The Constitution of India clearly states that the country shall have no one National Language, but will rather have a multitude of Official Languages so as to not elevate the status of one over the other. The one uniting language of all the states, as far as work goes, is English, because it IS the primary medium of instruction in majority of institutions across the country, and even if it is not, it is taught as a secondary language to be learnt.
Hope that answers your question