Here are the many ways this newer technology improves upon the central air conditioning system most of us are familiar with.
Energy/Cooling Efficiency
First off, the system itself is far more efficient than central AC in every situation. For example, if you have uninhabited rooms that currently don't need cooling, you can simply turn off that room's indoor unit. You only spend energy cooling the rooms that need to be cooled.
All you could do with central air conditioning is to shut the vent in that room. But the fan of the central AC's indoor unit will still be working just as hard. Plus, some amount of cooled air will still leak into that room through the shut vent, wasting energy.
Speaking of leaks, ducts are notorious for leaking energy, especially if they aren't properly insulated. Typical homes lose 20% of conditioned air to duct leaks.
But even with well-insulated ducts, the air loses some of its conditioned temperature by the time it gets to each room. This can't happen with ductless mini splits.
Furthermore, inhabitants in each room can choose their own temperature settings. Thus, they don't have to rely on one thermostat in a completely different room.
Ease of Installation
Mini-splits are much more convenient to install. There are no large ducts to worry about, only small tubes through small holes in the wall. Also, homes that have no room for ducts still have room for mini splits.