Flywheels and flexplates are two parts that help accomplish the same task in an automobile in very different ways. This depends on whether the driver can control the transmission manually.
Starting the ignition allows electrical energy to be converted into the mechanical energy of an engine. How the engine receives torque depends on the transmission and the accompanying flywheel or flexplate within it.
Either part is necessary depending on the type of vehicle. Flywheels are used in manual transmissions, and flexplates are used exclusively for automatics.
The flywheel is connected directly to the clutch, allowing torque to transfer between the transmission and the engine. The flywheel can dampen engine vibration and provide a surface for the clutch to contact.
The flexplate mounts the torque converter--which is a device that can multiply torque when there is a difference between rotational speeds--to the crankshaft.
A flywheel can vary greatly depending on whether it is light, so that it can spin quickly, or heavy, so that it has more inertia once it is spinning. A flexplate is typically stamped steel with a ring gear welded on.