Compared this again with the iPhone 5 earlier. No idea why people keep saying the S4s screen is too saturated, set to movie mode there is very little colour accuracy difference between the S4 and i5. The only difference you really notice is the S4s screen has higher contrast, better blacks, and is slightly sharper or cleaner. No black crush issue from the S3 either.
Whites aren't that far off imo either. They're slightly greener, but the i5s seem a bit off too, with a slight blue tinge. I will say, blacks have come a long way on lcd's, still not Amoled level but much better.
Speed wise, S4 transitions (at least on my settings) are generally quicker and snappier across the board, though i5s animations last a touch longer. Where the S4 pauses a split second to load intense apps, the i5 loads them, but pauses internally whilst stuff loads up. For example, the camera app loads instantly, but the shutter stays closed a split second till the app is ready for shooting.
Web browsing speed wise, they traded blows, i5 better on some, s4 on others. I'd say overall the edge went to the S4, especially loading sites such as The Verge where it was a second or more quicker to load. Loading more image heavy sites it seemed to have the edge too.
Aside from that I only tested general use. Didn't do a pixel peeping camera test but briefly comparing I'd say the S4s camera looked quite a degree more detailed, less noisy and more colour accurate (i5s a touch too warm).
In any case, Samsung's made several key strides here. Especially with the screen. Thinking of putting up a comparison video when I have the time.