My buddy was up at the Kawi shop in the 'couv a few weeks ago and the sales guy told him "yeah, those two are pretty much the same, they're like the same bike."
To us, that's crazy talk, but as Seeya's function chart illustrates, there is quite a bit of overlap. Looking at it from one of the extremes, they both suck as dirt bikes, but the Versys sucks more. They are both pathetically slow, but the KLR is more pathetic. Looking at it from the inside, we see a lot difference: Twin vs Single, Injected vs Carbed, 6-speed vs 5, Cartridge-fork vs Damper-Rod, 17" wheel vs 21", etcetera.
A sportbike guy, the Salesman went on to talk about the differences between the R1 and the ZX10R for twenty minutes. You could probably tell me about the features and benefits of a V-4 tourer versus an opposed-6. To me, They're both heavy, complicated, and expensive; not my cup of tea and why split hairs?
But then, I'm not looking for a touring bike. You mentioned single-track and off-road. There are a lot of better bikes for that than these two, but if you're just after a different riding-experience and want to dip a toe in the water, then a KLR would be a decent start.