Japan is where early air-source heat pumps first achieved market success, so it's unsurprising that they are much cheaper to install there, because of the relatively large number of installer options.
In the US, they are struggling to break out of the eco-luxury product niche (where they have been stuck for a long time).
Compared to a conventional A/C compressor (which they replace), heat pump compressors are much smaller, quieter, and less ugly.
As for the indoor units, they can either be the "ugly" ones (the indoor head units visible on the wall), recessed "cassettes", or they can use traditional A/C air handlers in a utility room to distribute conditioned air via existing duct-runs and registers.
There are also companies like Quilt that are making heat pump systems with much more attractive indoor wall units.
I was curious about sound for both the indoor and outdoor pieces. The outside condenser for my current AC is very loud. Are the heat pump units quieter, and if so, why?
Better variable frequency drives for both fans and compressor is a big part of it (see other comments about being less prone to short-cycling).
This isn't exactly new or unique to heat pumps (and some older heat pumps lack both), but as the technology has gotten cheaper and more reliable, coupled with the drive for better efficiency, it has become commonplace.
Speaking as an apartment dweller, my new apartment's ductless setup is much quieter (inside) than my old apartment's blower. At the old place, the fan was either on or off and must have added another 10 or 20 db of noise forcing you to always keep the TV remote nearby in case the fan turned on or off. It's very convenient that I can control my ductless unit's fan speed with a remote.
In the US, they are struggling to break out of the eco-luxury product niche (where they have been stuck for a long time).