What's the difference between a deep fryer and a chip pan?
The part that handles the food is comparable, as you noted: A container for the hot oil plus a basket to lift the food out again. 

But the difference is in the periphery: 

A chip pan is just a pan (or what you may also call a pot), but the heating is done on the stove, like for all other pots.

A deep fryer has its own heating system, either via a heated thick wire in the container or by heating the exterior walls. It’s an independent unit and actually shouldn’t be placed on a stove for various reasons. It typically has an insulated exterior and often a lid to minimize splattering and keep the oil from dirtying up the kitchen. If you see a built in fryer  in a commercial kitchen setup, it’s still a separate unit.