Simple answer is it’s down to the actual machine type. Eg, just because it’s an inverter doesn’t mean they will all be doing the same thing.
I’ll frame the response specifically talking about a high end inverter. Some of the features are reserved for only big industrial based equipment and some is a feature of almost all inverters in general.
The inverter can adjust the Parabolic curve, which is the relationship between Voltage and Amperage. Obviously the amperage relates directly to the thickness of material that you want to weld and is set by the operator, but it’s the voltage that keeps the electrode burning. For optimal results Each family of electrode (Basic, Rutile CEL etc) will require more/less voltage for the same amount current. So a good machine can/will adjust its output based on this.
They can also increase the arc force when the machine detects that the arc is getting too short as they have a large voltage reserve, so in extreme cases a good mark of a MMA machine is to establish the arc with a difficult to use electrode such as a Basic-Lo Hydrogen then plow the end of the rod directly into the weld pool and push the rod into the plate. A good machine will detect this and call on its voltage reserve and increase the welding volts to prevent the arc from freezing. So much so that you should be able to push a hole right through the plate.
Inverters also allow for a hot start. This isn’t possible with Transformer types.
You can also have a Pulse on the current. For many MMA applications this is no more than a gimmick but for welding very thin sheets and also for positional welding (vertical T Fillets) with a large electrode can be invaluable.
You can get a multitude of remote controls, simple thumb wheel types for on the fly adjustment to more complex ones. Some inverter machines have electronically controlled mechanical switches that allow internal change of the polarity of the dinse sockets. This means that the operator doesn’t have to return to the machine to switch over the electrode holder and the earth lead. This change can be done via the remote. There are also wireless remotes which allow you to dial in the welding value you want and then send this info to the machine via the copper cable in the electrode holder.
More practically, inverters can cope with long extension cables and can adjust their output to compensate for the input.