We actually have both, retrofitted about 5 years ago into a 1980s house. A Toshiba mini-split downstairs and a Carrier ducted system upstairs. The indoor units of our Toshiba mini-split are actually very quiet: barely audible unless you really crank up the fan speed.
In our case we couldn't install ducting downstairs (no space between the floors), but it was simple and cheap to do so upstairs because there was attic / roof cavity space in which to run the ducting.
A big advantage of the mini-split is that it gives you precise temperature control in each room, where as our ducted system only has a single central thermostat, and on/off control in each room. This is problematic for the bedrooms, which tend to overheat at night in the winter when bedroom doors are closed, while it remains cooler in the hallway where the thermostat is. I'm currently looking into an upgraded control system ("Airtouch 5") which should solve this by putting wireless thermostats in each room.
In our case we couldn't install ducting downstairs (no space between the floors), but it was simple and cheap to do so upstairs because there was attic / roof cavity space in which to run the ducting.
A big advantage of the mini-split is that it gives you precise temperature control in each room, where as our ducted system only has a single central thermostat, and on/off control in each room. This is problematic for the bedrooms, which tend to overheat at night in the winter when bedroom doors are closed, while it remains cooler in the hallway where the thermostat is. I'm currently looking into an upgraded control system ("Airtouch 5") which should solve this by putting wireless thermostats in each room.