Is it true? Yes, doing nothing uses less power than doing something. Displaying nothing uses less power than displaying images, which requires accessing the hard drive to get the images or using the GPU or CPU to generate 2D/3D images. Turning pixels on and off uses power. Not a lot, but more than nothing. Speakers need power.
What difference would you notice between a blank screensaver and just leaving the display on? Compared to just having the desktop displayed, probably a negligible difference in power consumption. Beyond that it depends on what application you're running.
Screensavers are for saving your screen, not conserving power. If you want to save on power disable the screensaver and set your display to shut off instead. Turning off the LCD backlight saves a lot more power.