Traffic!
Bari has a population of just under 320,000. During peak commuter times and holiday periods the 20-25 minute drive around the Baritangenziale can take up to 50 minutes, adding significantly to your journey time. The journey around Bari is unavoidable if you are heading to destinations south of the airport.
Brindisi has a population of just under 88,000. The traffic flow around the city is always much easier, even at peak times. If you are heading north from Salento airport you completely bypass Brindisi, for an easier, more relaxing drive.
If your first or last destination is Matera, the drive from/to Bari airport (via Modugno) is very straightforward and avoids the Bari tangenziale ring road.