As tempting as it is to attempt to circumvent the running vs. cycling argument by changing the comparison to something like "toned abs vs. spare tires," cycling does have some benefits. It allows someone with a spare tire to retain their shape while lowering their resting heart rate. Or something like that--the jury is out on the heart rate part.
In terms of heath benefits from a recreational perspective, running is superior to cycling in every way. The primary reason for this is the equipment involved in each activity: runners have shoes that they wear on their feet, whereas cycling involves sitting on a bicycle. It's much harder to coast or take a mile off when your feet are your only means of covering the distance. A guy who runs a lowly ten minute mile is responsible for every inch of that mile, whereas a cyclist can easily pedal just fast enough to remain upright. In that case, the tires work harder than the cyclist.
While this point is often countered with the contention that cycling involves a saddle rather than a seat--the difference being that one sits on a seat while a saddle is merely one of (usually) three points of contact with the bicycle, with the feet carrying most of the rider's weight--this contention doesn't often apply to recreational cyclists. Recreational cyclists sit and pedal. Their pedals don't often see as much pressure as their saddle. Blasphemy? Any cyclist--recreational or competitive--who wants to defend cycling from this perspective should try taking the saddle off of his bicycle before his next ride. Any reply may well come while recovering from a heart attack.
Another area where cycling is inferior to running is safety. When was the last time you saw someone running in a helmet? Given where things have gone on ski slopes, it may not be long before it becomes a more common sight. Regardless of this, though, cycling involves more speed and is an activity which...what's a diplomatic way to put this?...doesn't necessarily encourage a great deal of focus on the part of the rider--a combination which can lead to accidents as a result of carelessness, fatigue, or any other cause of distraction. When you're flying down a hill on bicycle, it isn't hard to imagine any number of things which can result in injuries ranging from cuts to broken bones--a level of injury which is extremely hard to achieve when running without the help of a car, or perhaps a collision with a cyclist who is flying down a hill.
Running burns more calories, stresses more muscles, and actually promotes better posture. A cyclist leans forward and holds himself up with his arms. Sore necks are normal, but are generally less sore than the area where the bike saddle (fine, we'll call it that) contacts the body. Which--again--is usually the result of sitting and pedaling rather then cycling. It's human nature to take notice of areas where it is easy to cheat, and cycling is far superior in terms of offering opportunities to do just that.
The downside of running, though, is exactly what makes it better: it's harder on the body. Every step is an impact, and unless one is at his optimal weight, any extra pounds are often felt in the feet, ankles, knees, and hips. It's like deciding between calibers of handguns--a 9mm handgun may have more stopping power than a .22 handgun, but that stopping power is only useful if you can handle the 9mm. It is from this perspective that cycling is superior to running--it doesn't matter if running is better than cycling if your body can't handle the impact and stress of running. Cycling is better in such cases because it removes the issues involved in running. The downside is that it takes a lot more cycling--both in terms of mileage and time--to accomplish the same level of benefits available through running. But the upside is extreme; if you can cycle in comfort but can't run without pain, it doesn't matter how much better running may be.
