Display is not off... the screen is not lit, but the display is not actually off.
It is in a power-down mode of some kind, similar as your Mac, sleep or standby, whatever.
As your Mac is sleeping, the display would need to be in a mode that allows a quick power up when you press a key, or whatever method you use to wake up your Mac.
The point I was trying to suggest was that the display may be on a different power scheme, depending on the computer that is in use.
I am curious, however... If you sleep your Mac, and note that the display's fan continues to run--then return to the computer later (maybe 30 minutes, or an hour), while the computer is still sleeping. Listen for the fan (don't touch anything, just listen for the fan). Is the fan still running after that time period? If the fan has shut itself off, then that's just how it works in standby mode when connected to a desktop Mac, and continues to a low power mode (fan off) more quickly on a laptop.
I guess that the difference is when there is a rechargeable battery in the power chain. The external display would likely power down more quickly (so no fan in sleep)
I don't know, I'm just guessing about this - Ask a "genius" at an Apple store, maybe you will get lucky and talk to someone that knows the right answer.