Overdrive tends to have a smoother, "sweeter" sound, found in a lot of 70's classic rock, like Neal Schon from Journey. A bit more modern example would be Slash from Gun's & Roses. Overdrive is often an accompanied by reverb and delay, a little or a lot, subtle or really noticeable, especially for solos. Pete Townsend from "The Who" would be a good example of overdrive being used without a lot of other effects.

Distortion has a more brittle, abrasive sound, found more in punk & metal. As PRSplaya has said, a more saturated sound, the idea being to really blend the notes together into one "crunchy" sound. A really good example of distortion is the guitar in "Creep" (I think the band is "Radio Head"?)... that really abrasive rhythm guitar that leads into the chorus. Nine Inch Nails is another good example of an over-all guitar sound that is based more on distortion than overdrive. Marylin Manson "Beautiful People" would be another distortion example. Distortion tends to be used without any other "spacial" effects like reverb or delay, but nothing says you can't use these effects, its just most guitarist using distortion are going for that dry, brittle sound.

You can probably get overdrive to sound more like distortion and distortion to sound more like overdrive, but it gives you and idea what the designers of the amp, pedal, rack or effect plugin had in mind when you see it labeled one or the other.