Anandtech posted an interesting article discussing the power consumption of Samsung's AMOLED displays, from the GS4 to the GS6. The basic results are show in the graph below, which compares the power consumption of a completely white screen at different brightness levels:
You can see that The GS4 is by far the worst performing screen of the bunch. Samsung is consistently improving its AMOLED displays, but from the GS5 onwards it also introduced PSR (Panel Self Refresh) functionality which helps reduce power consumption on static images.
Anandtech later isolates the screen themselves, summarizing the results using normalized values (mW/cd/cm2): The GS4 is still the highest at 3.41, the GS5 improves to 2.66 and the GS6 improves further to 2.21. It's great to see Samsung managing to increase power efficiency so quickly - while enhancing image quality (higher resolution, for example, draws more power, so this is a real improvement).
According to Anandtech, the latest LCDs outperform OLEDs by around 33% for white screens (or high-level grayscale images). Of course using a dark background has a dramatic effect on OLED's power consumption. The white-screen consumption gap is decreasing however, as OLEDs improve quicker than LCDs, and it's likely that OLEDs will outperform LCDs in this regarding in the near future