Gatwick is the natural choice for most of south London — direct trains connect to Clapham Junction, East Croydon and Victoria.
Gatwick is the best airport for south London. Southern and Thameslink trains connect Gatwick directly to Clapham Junction, East Croydon, London Bridge and Victoria — without needing to cross central London. Gatwick is geographically south of the city, making transfers to south London shorter and cheaper than Heathrow.
Here is how the airports compare for south London neighbourhoods:
Gatwick is south of London — the train lines run directly into south London stations. Southern trains serve Clapham Junction, East Croydon, Purley and Sutton. Thameslink trains serve London Bridge, Blackfriars, East Croydon and Brighton. You never need to cross central London to reach your south London destination from Gatwick.
Heathrow is better for southwest London — Richmond, Twickenham, Kingston. The X26 bus or a taxi from Heathrow can reach these areas faster than travelling from Gatwick. Also, if your flight only arrives at Heathrow, the Piccadilly Line to central London then a southbound train is workable (60–80 minutes total).
Gatwick. Southern trains take ~25 minutes from Gatwick to Clapham Junction. This is the fastest airport-to-south-London connection.
Heathrow is on the west side of London. To reach south London, you need to travel into central London and back out — typically 60–90 minutes. Gatwick is almost always the better choice for south London.