I own a Wee Strom. I've sat on a Versys but never ridden one. I love my Strom, but I'll try to speak objectively without gushing about how awesome it is.

From what I've read, both get about the same economy and have roughly the same torque/horsepower curves. Both are farily lightweight. And, both are pretty fricken ugly.

The V-Strom has more extensive wind protection, and factory handguards are available. The Versys's windscreen and fairing are pretty small by comparison. (Bear in mind that the Strom's wind protection is really just "okay." It keeps the wind off your chest, but the stock screen creates a jet of turbulent air that will hit you right in the face.)
The V-Strom is availabe with ABS. Hard to find, but available.
The Strom also has that beefy luggage rack, which a top case can bolt right up to. The Versys would require the purchase of a luggage rack, separate from the top case.
You can mount a centerstand on the Strom... the Versys's exhaust precludes this. That's a huge design issue, IMO.
Also, when I sat on the Versys, I noticed that the foot pegs were in a bad location for me. When I put my feet down on the ground, I was banging my shins on the metal footpegs. Ouch. A few other people have noticed this as well.
But...
The Versys does have a nicer suspension. Upside down forks mean less unsprung weight. The Strom's suspension is okay... but I'm really not as sensetive to that stuff as other riders here.
It also has wave brake rotors... which are, well, wavey?
In general, the Versys is virtually all streetbike, while the Strom can claim *some* off road capability. I rode mine on a muddy dirtbike course last weekend.

I do NOT recommend those stock Trailwing tires in the mud! I can't believe I didn't dump the thing.
It wouldn't surprise me if the Versys handled better, but the V-Strom is no slouch. I've had mine for 8 months now, and it still surprises me with how far it can be leaned into a turn. The mass of the bike is really centered and LOW... you really do forget how tall the thing is when you're riding it hard.
Hope that helps! As for me personally, I drank the V-Strom Kool-Aid and never looked back. It's one of the few things I've bought that truely lived up to its hype.

I'm sure the Versys is nice, but I personally would be looking more at the Ninja 650R if I were still shopping.
Rob