Geographically Lecce and the south are closer to Salento airport. Heading to Gargano and north of Bari will make more sense from Bari airport. But if your destination is the Valle d’Itria either might be suitable.
The most straightforward route to Alberobello is the same distance from both airports, although the drive from Bari airport takes longer.
Even the fastest route to Alberobello from Bari airport is slower than the journey from Salento airport.
Driving south from Bari airport takes you around the Bari tangenziale ring road- the busiest stretch of road in the Puglia. Heading north from Salento airport means bypassing the traffic around Brindisi.
It is also a much easier drive from Brindisi, giving you time to acclimatise to driving on the other side of the road should you need it.
Bari is the busier and older airport. Salento airport is newer and more compact, with a much easier in and out than Bari.