In the TM4C123G Launchpad Workshop page 129, there is a note about power modes. This note differentiate between processor clock and system clock. So what is the difference between them ? Is this a general concept or is it specific to the TM4C123GH6PM micro-controller ?
The processor clock is derived from the system clock, but for the mode you are mentioning it is just "disconnected" from the processor core. So the system clock is not stopped (in the Deep Sleep it is stopped as well), it is just stopping feeding the core.
From the datasheet, the processor has a main clock, which the system clock is derived from as shown in the image below:
Different clocks control different parts of the microcontroller. One of those clocks control the main code, while the other controls timers. By using low power modes, you turn off the main code clock but leave the other on so your timer or adc or other peripheral that runs on that timer still works. Essentially all microcontroller work this way.