I think this excerpt from an EZine article by Stephanie Hofstetler covers the differences pretty well. Personally... I LOVE LOVE LOVE Kettlebells and can't imagine going back to traditional weight workouts now. I still LOVE Cathe for her core work, stretching and other types of workouts, but dumbells are mostly a thing of the past for me now.
"A kettlebell workout differs significantly from free weight or dumbbell exercises. Although they have a lot of similarities with dumbbells, kettle bells engage all 38-core muscles because merely picking it up triggers instability since the center of gravity in a kettlebell exists on a vertical rather than a horizontal plane.
When you pick up a dumbbell, the handle is directly in line with the wrist. There is a direct line of what is called "Applied Force." This is because the dumbbell is symmetrical due to its balanced grip.
However, when you lift the kettle bell by the handle from the floor, you are engaged in a pulling motion. As you raise the kettlebell, its center of gravity relative to your body begins to shift. At approximately shoulder level, the pull on your bicep, tricep and shoulder muscles becomes a pushing motion. The kettle bell's effectiveness is in its instability since its very design is asymmetrical. There is no direct line of applied force.
The effect of this change in the center of gravity in this sample exercise is to engage stabilizing muscles at multiple points in and around your arm and core. By contrast, a standard biceps curl engages only those stabilizing muscles necessary to support your arm in a single plane of motion.
Another difference between kettlebells and dumbbells is the versatility a kettle bell provides. You can do a total body workout using a kettlebell in 20 minutes including cardio and resistance training. You cannot do that with a dumbbell. This is due to the kettle bell's unique shape.
Some of the exercises you can do with a kettle bell included power swings, power squats, snatches, cleans and Turkish get-ups. These you have to see to understand. There are several videos on YouTube demonstrating the various kettlebell exercises. "