I know the question is about road bikes, but I submit some of my experience as MTB rider hoping to trow a little light on the subject.
As a mountain biker who has used a variety of components ranging from lower end cheap products to middle range stuff, I have noticed some differences in wheels, but especially in technical trails, rock gardens and the like.
In the case of mountain biking, wheel rigidity helps maneuverability, and the rider gets a more precise feeling and at least, the sensation of a faster bike reaction to rider's input. A more rigid wheel assembly also gives the rider a lot better feedback about the bike's interaction with the terrain.
In mountain biking, wheel flex has the effect of fading the smallest vibrations produced in the tire-ground interface. This effect is further amplified trough suspension components, so it gets more difficult to "feel" when a tire is about to skid or to determine if you can get faster over the current ground type. Such kind of assembly can also introduce estrange vibrations thus causing incorrect feedback. (I used a wheelset that constantly gave me the false sensation of rolling on a flat tire, even if the tire was properly or overinflated)
As for pedaling, I noticed a little side to side flex when climbing the steepest sections.
Another aspect of it, is that a flexing wheel, specially a cheap one, is more prone to material fatigue and failure. Wheel flex is a result of the quality in the materials used and the labor qualifications while assembling, so even high end materials can be turned into a under performing wheel by inexpert or lousy labor.
A properly assembled wheel will perform correctly and also will need to be trued less often. (For example, a guy once assembled a XC wheel set for me, using aluminum rims and 36 common steel spokes, which I abused with downhill use, including 3+ feet drops and very long loose rock sections, and they only had needed to be trued once after 3 years of use, and I really could tell whether they needed it because the bike uses V-brakes!)
As for road bikes, I have little experience, but I guess most of this still applies.