In the context of machines or systems such as heating systems, computers, etc, the difference is that a person turns a machine on or off, but a machine can come on or go off by itself (e.g. if it has a time switch). Electricity in a house can go off if there is a power failure and come (back) on again later.
To turn something on in this context means to make it work by pressing a switch, pulling a lever, etc.
Definition of turn on
1: to activate or cause to flow, operate, or function by or as if by turning a control
turn the water on full
turn on the power
Turn on (Merriam Webster)
In this context also, if something
comes on, it starts working
to start to happen or work
The heating comes on at six in the morning
Come on (Cambridge Dictionary)