There is enough info on the KLR around here, just do some searching, so I'll focus on the Versys.
Just looking at the Versys I cannot see how anyone could consider it a dual sport. Kawi hasn't said anything to insinuate this either. They have it in the dual purpose menu just because it's close enough. It's being touted as a "multi purpose" motorcyle. Multi purpose being practical commuter, urban assault vehicle, Sportbike and Light Tourer. They say nothing about expecting people to take it offroading. A little bit of excellent condition forest roads can be ridden on any bike, that doesn't make them dual sports.
I don't know how tall you are, but the claims about being comfortable have to be BS! Take a look at this guy...
Versus this guy
The first photo makes the bike look pretty small which it really is, where the second photo makes it look larger. 3" shorter wheelbase then the KLR. 2" shorter seat height. That's quite a difference to those who already feel a little cramped on the KLR.
Only you can decide what your willing to put up with. Nobody can really tell you what will make the best bike for your needs. You just need to read all the specs and go look at them in person.
EDIT: OH YEAH.... don't forget the muffler location on the Versys! That bottom plate is for minor debris coming off the tire from the front, it doesn't cover the muffler. Talk about a bad day when a log or rock smashes the piss out of the muffler can. Or if you end up in some ruts and the back tire drops down and smashes the muffler can on one of the dirt humps in between the ruts. Same could happen for deeper potholes that are normally found offroad. Nope... this bike isn't going to survive anything harsh. I've had the back tire of the KLR in the exact position I'm talking about many many many times and I guarentee that muffler on the versys would have been busted all to hell.