As an owner of both 650c and 700c-wheeled bikes will happily share my thoughts.
First of all, 650c wheels are lighter and obviously smaller which makes the whole ride more dynamic which can improve your maneuverability.
On the other hand, 650c wheels are prone to sliding on stones or corners so you must be more careful when riding. 650c wheels spin faster so be ready to change tires sooner than on 700c bike.
Most makers make good tires for both 650 and 700 wheels, but 700 are more popular and therefore you end up with better choice. When it comes to wheels the picture is even grimmer... However, it just takes little more searching - many makers produce 650c versions of their wheels too.
The size difference between 700 and 650c wheels is very significant so there's no way to fit 700c tires onto 650c wheels.
Changing wheels from 700c to 650c is in usually bad idea, because you will have to move brakes (usually impossible) and most likely the bottom part of your bracket will be too low and pedals too close to the ground to use them.