What applications require isolated transformers?
The problem with mains power is a dangerously high voltage that is referenced to ground. That means that if you are standing with your bare feet on your floor and touch (the wrong) one of the mains conductors, you will get an electric shock. These shocks can easily be lethal. Now the problem with an auto-transformer is that you can never be sure that you attach the common wire to the neutral side of your mains supply. Therefore depending on how you plug your auto-transformer into the wall socket, you may get a lethal zap when you accidentally touch its output, 50% chance.







simulate this circuit &ndash; Schematic created using CircuitLab

By using a proper transformer, the secondary side is galvanic separated from mains power. That means no matter which end of the secondary winding you touch, you won't feel a thing. Only when the output voltage of the transformer is high (say > 40V) then you will feel it, but only when you touch both wires at the same time.

Medical applications as mentioned in other answers are on the extreme end of why you want galvanic separation from the mains supply. Medical applications require a fair amount more precautions than just a simple transformer. These transformers typically have extremely low capacitive coupling too, but that is worth a question on its own.