Difference between current transformers and power transformers
A transformer is a transformer whether intended for current sensing use or power conversion use.  All transformers work on the same principle.

However, there is considerable latitude in various parameters when designing a transformer.  These different tradeoffs give the transformer different characteristics and therefore make it suitable for different applications.

A current sense transformer is optimized to have small primary impedance so as to minimize the voltage drop in the line it is intended to measure current in.  The secondary is also intended to be connected to a low resistance.  This reflects a lower impedance to the primary.  The transformer is run primarily in short-circuit output mode.  Note that little power is transferred thru the transformer.  Energy is taken from the magnetic field by the secondary almost as soon as it is put there by the primary.  As a result, the core can be small since it never has to hold much energy at any one time.

A power transformer has a different purpose, which is to transfer power from the primary to the secondary.  Sometimes they are just for isolation, but often it is also to get a different combination of voltage and current on the output than the input.  To get power, you need both voltage and current, which means the transformer needs to be operated somehwere between short circuit output where there is no voltage and open circuit output where there is no current.  Generally power transformers are designed so that the secondary looks reasonably low impedance and therefore it's voltage doesn't sag too much at the rated power output.  They also have to behave reasonably with light load or no load, meaning the open circuit case.  Again you want low impedance so that the voltage in the light load case is not too different from the full load case.  This type of transformer has to be able to handle larger energy in the magnetic field.  This means a physically larger and therfore heavier core.