For one thing, while bright sun can create interesting shadows and lighting, it can also create a lot of blown out "hot spots" that are often difficult or impossible to correct later. Any extreme in a photograph, light or dark, is bad because your camera is not recording information/detail in those parts of the image. Even if you plan to use Photoshop or another editing software, if the info isn't there, the program can't help much.
Shooting on an overcast day there will be virtually no glare from the sun or dark, unwanted shadowing. This is something to especially consider if you're planning on doing any nature close-ups like flowers, leaves, insects, etc.
Grasshopper! Concord, NH, 2009
Another overcast perk is the clouds. Clouds can provide a lot of interesting textures and contrast to your picture, even if you can't tell right away. I shot this picture on a cloudy day and while these clouds looked fairly non-threatening in person, a quick simple contrast adjustment created a dramatic difference.
Jamaica Pond, Jamaica Plain, MA, 2010
Same with this picture shot on a fairly "flat" day-
Solitude, Madison, NH, 2007
Don't forget too, if you really like the blown out/high contrast look, it's always better to get a good shot with as much information as possible and work with it later when editing. This will give you more options/control over your image and what you want to do with it!