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Flywheel is for a manual transmission.
Flexplate is for an automatic.
Flywheel has a flat surface (looks similar to a disc brake) for the clutch disc.
Flexplate looks like a piece of 1/8" thick sheet metal with holes in it.
Both can be balanced or not.
Both have teeth around the outside edge that the starter uses to turn the motor over.
Flywheels tend to be fairly heavy (40 lbs + unless you get an aluminum flywheel).