There are tons of advantages and disadvantages, but i mention only the relevant cons and pros.
Indirect (belt) drive usually produces less noise in compare to direct drive (continue power transmission vs discrete power transmission like gears and chains) but you know engineers had done a lot to reduce the noise in household equipments, unfortunately not yet in industrial sector.
The belts are far more prone to relaxation and then they can transmit less torque. As you might have noticed after some years after you replace the belt. Again this is less observable for home and kitchen equipments.
The belt drive damps the vibrations, the direct drive produces more vibration specially as the rpm goes up, but in washing machines the direction of rotation changes time to time and the direct drive offers more stability.
The maintenance cost of belt drive is lower than direct drive, easy to disengage for repairing.
The belt drive applies greater load on the main shaft of the machine, it has a direct effect on the life time of bearings, shaft misalignments and as a results oil leakage and more vibrations.
The direct drive requires less space in compare to belt drive so smaller machines.
The belts are sensitive to the surroundings temperature, humidity, oil ... . So the probability of failure is higher plus the micro slip.
The inverter is a sort of speed, and power control over electric motors, long story short it makes the rotation of the induction motors more smoother in compare to classic motors without inverter, of course it costs the clients more but it also regulate the power consumption.