There are tons of advantages and disadvantages, but I will mention only the relevant cons and pros.
Indirect (belt) drive usually produces less noise in comparison 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 equipment, unfortunately not yet in industrial sector.
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 equipment.
Belt drives dampen vibrations, direct drives produce more vibration specially when the rpm goes up. But in washing machines, the direction of rotation changes time to time and direct drive offers more stability.
The maintenance cost of belt drives is lower than that for direct drive, they are easier to disengage for repairing.
Belt drives apply greater load on the main shaft of the machine, it has a direct effect on the life time of bearings, shaft misalignment and as a result, oil leakage and more vibrations.
Direct drives requires less space in comparison to belt drives, so smaller machines.
Belts are sensitive to the surrounding temperature, humidity, oil ..., so the probability of failure is higher plus the micro slip.
Inverters are a sort of speed and power control over electric motors, long story short, they make the rotation of induction motors smoother in comparison to classic motors without inverters, of course they cost the clients more, but it also regulates power consumption.