Unlike North India, South India’s foundation of this festival is on the bases of Mahabharata , not Ramayana.
In the South, Diwali is about Asura (demon) called Narakasura being killed by Krishna. That is why Diwali is widely known and celebrated as Naraka Chaturdasi. In South, the festival Naraka Chaturdasi is celebrated one day before Diwali, which falls on Amavas and considered to the actual start of the festival. This day, children wear new clothes, people exchange sweets and visit their relatives. In Tamil Nadu, people generally burn crackers on both the days, while in Karnataka and Andhra Pradesh, Naraka Chaturdasi is usually associated with oil baths, house cleaning, sweets preparation.
Laxmi Puja is common in North and South India. In South, it is celebrated day next to Naraka Chaturdasi. It may fall on Amavasya or on the New Moon Day. This is day when families worship Goddess Lakshmi, for prosperity and fortune, light up their homes with diyas. And most of the crackers are actually burnt on this day, though in Tamil Nadu, people burn them on both the days.
So, Diwali in South is essentially Naraka Chaturdasi, followed by Lakshmi Puja, and that makes it seem a day earlier.