The biggest difference is that the Konis are only rebound adjustable while w/ the KYBs you adjust rebound and compression with one knob.
So adjusting the KYBs will have a bigger effect on chassis balance and ride, but the Konis can be tuned finer to different springs. W/ the KYBs you get more rebound AND more compression damping when stiffening them. So if you put on a stiffer spring you'll need more rebound damping but the associated increase in compression damping may be the opposite of what you want.
If you're looking for a strut that you can fine-tuned for all-around and leave it, you might want to go for the Konis. Then get a spring with a rate that will give you the ride stiffness you desire.
If you plan on adjusting it a lot, the KYB may be the way to go. KYB would be more attractive to me if the rears were 7-way adjustable like the GC8 units. That would give you finer control over chassis balance. The Konis, on the other hand have a very wide range of infinite adjustability for rebound damping.
First off I think that the Konis are more expensive than the KYBs. I haven't seen Konis for less than about $550. Like Big Sky said Konis are more popular for competition. They also have the advantage of finer adjustments in damping and if you're serious you can get them custom valved. I'm actually considering a Koni/Ground Control setup but at that point I have to wonder if I should just go to a "real" coilover setup.