The difference is that a current traveling through a circuit flows constantly through conductors, typically in a low voltage but at a high rate of current. A static discharge is high voltage but low current, and occurs due to a build-up of opposite charges on objects separated by an insulator. This build-up occurs until the charge is so great that the electrons flow though the insulator to balance the charges.
Static discharges can be very harmful depending on the medium that the charges are traveling through and the objects that the charges are transferring through. Discharges that occur through air results in a visible shock as the electrons from the charge reacts with air releasing energy in the form of light, such as what occurs with lightning. Circuits require a closed, never ending loop of a conductive material; they wouldn't work if there were any insulating gaps such as air. The charges constantly flow through the conductive material in the same direction, never accumulating or drastically changing.
Electronic components rely on circuit electricity to work and can be damaged by the spikes of charge that occurs with static discharges. This is why special components in circuits, such as fuses or breakers, are used to protect them.