Maybe it's a difference in production batches? Every test I've found online has the Note 2 at half the brightness of the iPhone 5 though. The only screen I've found to come even close to the iPhone is that of the One X, with the Nexus 4 being at least a bit brighter than the Note 2.
This plays into the AMOLED vs LCD technology difference. LCD has better brightness, but AMOLED get's deeper blacks.
On most LCD devices (I know it happens a lot with iPads, and it happens on my Nexus 10) when there is a completely black screen, you'll get light bleed from the sides and/or corners. No such issue exists with AMOLED panels - blacks are as black as the ace of spades.
That said, there are improvements being made in AMOLED displays that improve the brightness.
The reality of it is that LCD has been developed pretty much as far as it can be. There is nowhere else for LCDs to go.
AMOLED displays are being tuned and refined with each iteration. An example is how the S4 has the feature of being able to change the contrast and saturation of it's display to suit what the user wants. You can have the bright, 'popping' colours of deep saturation, or the cooler, more realistic pallette offered by the more realistic tuning.
It still may not be as realistic as LCD displays yet, but it's getting there.