Think of the BGE like a Toyota; cheap, reliable, ubiquitous and it gets you where you need to go. The KK is like a Maybach; an elite thing of beauty that looks great and gets you where you want to go in comfort and luxury. A BGE is a very nice ceramic cooker and has the benefit of a nationwide (worldwide?) chain of stores that can provide parts and service. A KK, on the other hand, has a single point of contact that you can order whatever you need and you've got a much more personal relationship with the company. A KK also has much better insulating properties which means that they're more efficient at keeping the temperature constant with less coal. Finally, Big Green Egg made their name by making an ugly cooker in whatever color you wanted (as long as it was green) that stands out at BBQs but performs very well. KK's are much easier on the eyes and perform even better
Both cook fairly well with the KK's more even heat and more efficient thermals providing a slight edge, but KK is a much more friendly and personal company and looks much better on your patio.