Rim/Wheel stiffness: please explain

Stiffness is a really hard thing to quantify. For me, stiffness manifests itself as cornering confidence. I actually have far more cornering grip on a stiff setup.
But the only way I would be able to compare 2 sets of wheels for stiffness would be to ride them on the same bike, with the same setup, in the same conditions and see which one cornered better.

I found putting stiffer wheels on my bike to be like going up to a thru axle or putting on a stiffer thomson stem. Same feeling of more conection to the ground.

As Alias put it....

it is very difficult to quantify. It's one of those things that you feel more than anything else. A stiffer wheel will feel more solid in corners, and will accelerate more efficiently due to less spoke wind up. But to actually experience it or notice it you have to have ridden a fairly flexy set of wheels before. But contrawise a wheel that is too stiff can be unpleasant to ride, particularly on a road, ridged, or hardtail bike. So a good, reasonably stiff wheel will corner better and accelerate better. Just don't go overboard and end up with a DH or Freeride wheel on your XC/Trail bike. Overkill like that usually ends up negating the advantages of a stiffer wheel through increased weight and a harsh ride.

Good Dirt

Here is a good comparison.

My road bike came equipped with a set of Alex ALX200 wheels. Now I'm 6'2" 190lbs in fairly good shape. On sprints and steep climbs, the rim braking surface would rub on the brake pads. Hence flexing the wheels.

Now I built up a new wheel set - DT Swiss 1.1 rims, DT Swiss Comp DB spokes and Chris King hubs in a 28 spoke count. Now this wheelset is almost a full 1lb lighter than the Alex wheels, but I have zero flexing of the wheels when sprinting.

Wheel stiffness - Lateral deflection of the rim off axle center. This is why mavic loves their flat spokes - supposed to be laterally stiffer with force applied.