What is the difference between adding two drone batteries in series vs a parallel connection for flying the drone?
Batteries in series = add the voltages, ie two 3S 1300 30C packs becomes 6s 1300 30C.
Batteries in parallel = add the capacities, as above you get 3S 2600 60C.
There's no particular pros or cons for either, many larger commercial drones use combinations of packs in series and parallel to get the power source they need.
For smaller quads though the overhead you get in extra weight from the leads, connectors and wiring is greater so it's generally less practical.
As is the case with all other kinds of batteries (voltage sources in general), connecting batteries in parallel increases the maximum current which can be supplied to the motors and a series connection increases the voltage supplied to the motors.
I wouldn't recommend creating either circuit on a drone with individual battery packs on a 5" class or similar quad.