Another difference not mentioned here is accuracy. If you use a potentiometer you are using it as a ratio between two values. Like a voltage divider. Its absolute value tolerance may not be very good, but if you only care about the ratio set by the wiper, the pot is much more consistent over many devices and temperature.
But if a rheostat configuration is used, say in series to vary the current through a load, its absolute value is used. Not as consistent.
This applies to digital pots as well. See any datasheet to see error curves between the two.
For example this is the more accurate way to use a pot for variable opamp gain.
Below is not the way to get the best accuracy from a variable gain amp.
It's using the absolute value of the pot not the ratio set by the wiper.