Very little, in the main 12 volt used as with bathrooms 12 volt is the regulation limit in some areas, so if going extra low voltage may as well go to a voltage which can be used anywhere.
An LED is current dependent not voltage, there may be around 3 volt across a white LED, but it is the current that matters, so we use a driver, which will supply a set current DC to the LED, the simply driver may be no more than a resistor, so 3 white LED's and a resistor in series placed on a 12 volt supply. But clearly the resistor produces no light, just heat, so although the LED may produce 100 lumen or more per watt, as a package with the resistor it may drop to 75 lumen per watt.
So if we use a pulse width modulated driver, we lift the lumen per watt back to 100, so bulbs designed for caravans and boats often have a voltage range of 10 - 30 volt because inside the bulb is a PWM driver, which also means more light per watt.
It does however seem a little daft to have power supply feeding a power supply, if the LED is rated 340 mA then having a 340 mA driver seems the way to go, not a 24 volt power supply feeding a 340 mA driver. So you get a 15W 340mA Constant Current LED Driver, 10-44Vdc which is designed to run the LED direct. As we use different colours of LED so the volts change, so red LED around 1.2 volt.
So I bought some Lidi LED strip, it has an output of between 2 and 22 watt, and it has tricolour plus two white LED's one warm the other with high colour temperature, and the controller does not permitted you to use them all at max output, seem to remember a 6 wire control, and the controller is matched to the LED strip. I use it both under counter lights and display cabinet lights, with latter set to minimum of 2 watt. I would not dream of trying to change the power supply or zigbee controller, I have bought it as a package, and fact than strip will unplug is really only to make it easier to install.