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What is the principle of the difference between a supply which provides 0-18V and +9V and -9V?
If I am using a circuit which takes a +/-9V power supply, and instead start powering it with 2 9v batteries, is there a fundamental difference in the supply, or is it just a verbal one?
In this case BAT1 has at the plus side 18 V, and at the BAT2 minus side 0 V, because GND is connected there.
simulate this circuit – Schematic created using CircuitLab
Below, the GND is in between the batteries, so on either side the difference is 9V, resulting in +9V at the + side of BAT1 and -9V at the - side of BAT2.
Inherent in your description of +9 V and -9 V is a presumed 0 V connection. So you have two supplies: +9 V and -9 V
If you don't have this 0 V connection in any form, then your +9 V and -9 V can be renamed 0 V and 18 V. Technically, you could name them -2 V and +16 V, or -100 V and -82 V if you want. But this would be unconventional and very bad practice because it would mislead others into expecting a 0 V connection that they are referenced to. Keep it simple and conventional, your work has to be understood by others.
A supply that provides 0 volts and +18 volts can be assumed to have only two connections. A supply that provides +9 volts and -9 volts can be assumed to have a centre (0 volts) connection making it much easier to use in some op-amp applications.
