Why is a potential divider not an auto transformer?

Potential Divider

A voltage is a difference in potential between two points.

A potential divider is simply a pair of resistors in series or a variable resistor used to provide a split point between the two end points of a voltage applied across them.

As such, it is able to provide a fixed or variable potential at its center point or at the wiper of the potentiometer.

Power is not conserved in a circuit of this type, the current of the new output voltage is lower than the input current.

Transformer

A transformer, converts electrical energy whilst maintaining (within practical limits; there are some small losses) the same power ratio - i.e. the Voltage and Current entering are transformed to maintain the power ratio at the output.

This is done by having concentric, insulated, coils of wire around a shared metallic core. The number of windings in each limb of the transformer is used to determine the transfer ratio, e.g. (the following numbers of windings are ridiculously low to keep the math simple)

100 windings on the input side and 50 windings on the output side makes the transfer ratio 2:1, so 240 volts in will give you 120 volts out at double the current

100 windings on each side makes the transfer ratio 1:1, so 240 volts in gives you 240 volts out at the same current

50 windings on the input side and 100 windings on the output side makes the transfer ratio 1:2, so 240 volts in will give you 480 volts out at half the current.

Autotransformer

An autotransformer, unlike normal Transformers, provides a single winding with multiple output terminals, i.e. it is a single coil with two endpoints and several intermediate points, from which power can be tapped. This corresponds with its name: auto (self)-transformer

It operates by applying the input voltage to any two terminals and tapping the output from any two terminals, provided that one is a common terminal with the input.

Autotransformer vs Potential Divider

An autotransformer converts voltage from one level to another whilst maintaining the same power, i.e., voltage x current = P at the input is converted to new voltage x new current = P: the power remains the same.

A potential divider on the other hand, creates a new potential at its center/wiper. The new, lower potential is also at a lower current, the power is reduced.