Aerobic system simply means air is injected into the tank to encourage the growth of aerobic bacteria instead of anaerobic bacteria.
This system will likely have at least 3 tanks, a pre-treatment tank, the aerobic tank where air is injected, and then a holding tank for treated effluent.
The holding tank will have a pump that sends it to sprinkler heads. So you do have a leach field of sorts, the field where the sprinklers water.
There is obviously some risk here, the effluent is not fit for drinking, so pets, etc. could get sick.
Advantage? the "leach field" is all above ground and visible, no tree roots to enter the drainage tiles, etc.
I think the downside is obvious, but there is upside, unless regulations bite you in the butt. Regular maintenance and monitoring will be CRITICAL.
Also, obviously this type of system isn't possible where there is a deep freeze risk. I assume it's a southern state.