wow, but please tell us, what did you do while keeping the screen on? (beside that 12 minutes call
)
On topic two examples: the S4's battery is about 66% bigger than the iphone 5S. What we certainly know is that the display of the iphone is more power efficient (lower resolution, smaller display) which eats up the advantage of the OLED Display while displaying black pixels. You can assume that the iphone's SoC is also more power efficient since there are less cores and the clock is lower.
So you have a smaller battery but also more power efficient parts which leads to a similar performance on both devices (Maybe with a little advantage to the iphone).
BUT, things like cellular data or wifi eat up the same amount of energy as there is no big difference between those chips. So for example an hour of using the LTE chip and antenna uses relatively more energy on the iPhone than on the S4.
So far about the theory; The performance of both devices is very dependent of the usage you are doing. In fact both devices are capable of bringing you through the day in normal condition, so I'd just compare the overall experience since there is not really a difference on the battery side