I had a KLR.
If you haven't ridden both, I can tell you it's like comparing a horse-drawn cart to a jet. Seriously - they are nothing alike, IMO.
The KLR is a thumper, ~30hp, carburated, released in 1987 or so. Cheaply built, prone to a few issues (doohickey, subframe bolts, probably a few other little glitches I'm not remembering). More dirt-oriented. Crap on the highway, IMO.
The Wee (I'm a little out of my element here - I own a Vee) is a V-twin, ~60hp, fuel-injected, released in 2004. Don't know of any "known" issues on the Wee. More street oriented.
If fuel economy is your main concern, then there's little more to consider - but I think your riding style is what will really lean you towards one of the other. If you want to be more comfortable at speed, and value the convenience of a gas gauge, a clock, etc. - get the Wee. If you'd rather knock the thing around in the woods, or don't really plan on ever going for extended trips or on the highway, and you really want a simple, cheap, practical machine, it's hard to beat the KLR.
The ergos, CG, and handling characteristics are pretty similar, IMO.