The Edge adds nothing but the fact that it looks pretty. The physical bend in the screen has no purpose itself. It's all software based stuff for "swipe in from the left or right", but there's no reason they couldn't implement that software on a phone with a flat screen.
The only thing it actually has over competitors is that you can see by a coloured stripe who is calling you if you lay your phone face down on a table. But honestly, if you lay your £600 phone down on its screen you deserve a kick in the nads. There's no scenario where you can't just put the phone on the table right side up....
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only feature the Edge adds that's actually useful imho is the battery charge indicator. But then again Motorolas with amoled screens do that and so does Lumia with Glance.
It's such a shame that besides the Edge, Samsung has gotten completely stuck on features. They design their own SoC, but in the end it ends up having all the same stuff as the high end Snapdragons. Then Sony focusses on cameras, DACs, having a compact version that's still equally fast as the big one. LG is adding buttons in new places and making modules to stick in their phone, Huawei has incredibly aggressive pricing and pretty awesome service. And then there's Samsung, who don't really do anything innovative anymore