DisplayMate, the experts on display optimization, testing and evaluation, posted a new article comparing the GS4 display (4.99" Full-HD, 441 PPI) to the iPhone 5's LCD and the GS3. The GS4's Super AMOLED is a "very impressive display", and DisplayMate says that OLEDs have now reached the stage when they challenge the best LCD displays. There are advantages and disadvantages to both technologies, but we have to keep in mind that OLEDs are advancing at a much more rapid rate compared to LCDs which is now a mature technology.
Compared to the S3, the S4 is brighter by 25% (and when using Automatic Brightness, it's 68% brighter as this setting allows the brightness to reach a high level unreachable in manual brightness mode), it's 20% more power efficient (due to the green PHOLED materials) and it's got 44% more pixels per inch (and more than double the total number of pixels).
Compared to the iPhone 5, the S4 display is less bright. The iPhone's LCD has a bit better color calibration, although the white is more accurate on the S4. Of course the S4 display is much larger and have a higher resolution. The blacks are blacker on the S4 and the screen is more uniform. Overall - DisplayMate say this is a tie.
DisplayMate says that even though the S4 is a PenTile display, due to the display's very high resolution, it is sharp across the board. In fact, as they point out, there are 312 red and blue sub-pixels per inch - almost the same number of pixels in Apple's Retina displays (these are 326 PPI).
One more interesting point - it seems that the S4 has very low screen reflectance (in fact DisplayMate says it's got one of the smallest values it ever tested) - and this means that the display performs great in high ambient lighting - even though it's not very bright compared to the iPhone 5 display (the S4 is brighter though than most LCD smartphones).