There are some advantages to the 650. Smaller rim = lighter weight, less rotational weight, greater acceleration, overall height of rim with tire is less than equivalent 700c, overall height of bike lower, and many say the handling is better with the smaller tire rim combo.
Negatives are: harder to find tires, must order almost exclusively on internet(expensive) and carry spares around with you, with few exceptions you will need a custom built frame which will limit you to steel (if you care about material choices), and are generally associated with retro type designs.
Cant think of any more at the moment. Jan Heine seems to be the biggest proponent and promoter of this.