I am reading that the principal disadvantage of autotransformers is that unlike ordinary transformers, there is a direct physical connection between the primary and the secondary circuits, so the electrical isolation of the two sides is lost. What applications that use a regular transformer require the high and low windings to be electrically insulated from one another? Thanks!
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.
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.
The one that springs to mind is "any medical applications". When electrical equipment is used in a medical setting (or any other setting where electrical safety is a priority), an isolation transformer is considered an essential (and legally required) part of the power supply chain. This significantly reduces the risk of electrical shock.
Adequate isolation between a power source and a user of electronic equipment ensures the safety of that equipment. Given the high voltages that exist in modern electronic equipment, proper isolation protects an operator from contact with excessive electrical energy should a short circuit occur in the equipment. Isolation transformers have represented a traditional solution for providing high isolation in electronic circuitry. Even with the increased use of efficient, switched-mode power supplies (SMPS), isolation transformers can improve the overall isolation of an electronic design without severe penalties in added size, weight, and cost.
Isolation transformers offer an effective means of meeting the requirements of domestic and international safety standards for electronic equipment. In the United States, for example, such standards are set by the Occupational Safety and Health Administration (OSHA), with product testing performed according to appointed laboratories, such as Underwriters Laboratories (UL). Throughout Europe, safety standards are established by the International Electrotechnical Commission (IEC), with testing performed by the laboratories of individual member nations, such as the Verband Deutscher Electrotechniker (VDE) in Germany.
Isolation transformers enable a variety of electronic systems to meet safety requirements. Such systems include medical diagnostic equipment, computer systems, and telecommunications equipment. The systems may incorporate linear power supplies, SMPS, and sometimes a combination of both. A single isolation transformer can help an electronic design meet all of its isolation requirements. With proper system design, an isolation transformer can also help reduce the size and cost of the power-electronics components following it in a design.
Another widespread usage of Isolation Transformers is in marine applications and especially steel hulled vessels. In this application there are other considerations apart from safety the greatest of which is the mitigation of galvanic errosion of underwater metal parts when connected to shore power.
Reference: BS EN ISO 13297:2012: Small craft. Electrical systems. Alternating current installations
Use a transformer to isolate, applied throughout the circuit in which interaction between parties is required, without offering a common path to power.
In medical applications, the patient connection to the main supply line through the ground wire, can be deadly, even for very low current levels.
In other cases, is to couple a signal to a power line, such as PLC (Power Line Communications).