I hope this helps for those not familiar with transformer terminology.
The most common type of transformer has a winding for the mains & the output voltages are separate windings or separate windings with various taps on those windings.
That gives isolation between the mains & the output voltages.
An Autotransformer has a single winding.
One end of the winding is a common for the mains & the output for the neutral or cold connection.
For it to be a step up transformer then the other mains connection (active or hot) is a tap part way up the winding & the output voltage is at the far end.
For a step down transformer, one end of the winding is still a common for the mains but the the output is the tap part way along while the other mains connection (active or hot) is at the far end of the winding.
Autotransformers are cheaper to produce because they use less winding ie copper & possibly less iron as well.
Autotransformers have two disadvantages.
1. There is no isolation between the mains input & the output because they share the winding.
2. If there is winding fault then the output can be directly connected to the mains input & receive the mains voltage.
If the autotransformer is stepping the voltage up then the output device will be running on a lower voltage but if the autotransformer is stepping the voltage down then the output will get over voltage. The fault may be a winding failure but even just a bad screw connection or a broken wire. An example of the major problems this can cause is if a non-USA person is using an autotransformer stepping down 220V or 240V to 110V for a USA appliance then the failure will cause the 110V appliance to cop the full 220V or 240V & this may even spectacularly destroy it.