The difference between a flywheel and a flexplate is a flywheel is thicker than a flexplate and a flywheel is used in manual transmission engines while a flexplate is thinner and used in automatic transmissions.
The flexplate is a type of flywheel with teeth for the starter like a regular flywheel but it's also thinner and can flex hence the name flex plate.
Flexplates are generally much thinner and lighter than flywheels due to the smooth coupling action of the torque converter and the elimination of the clutch surface.
Like flywheels, flexplates normally handle coupling to the starter motor, via teeth cut along the outer edge of the plate.
The flywheel is found primarily in vehicles that have a manual transmission, whereas the flexplate is found in automatic transmission vehicles.
Both of these components are typically heavy, but the flexplate is much thinner and therefore lighter than the flywheel.
While there are some exceptions, if your vehicle has a manual transmission, you want a flywheel, and if your vehicle has an automatic transmission, you want a flexplate.
A flywheel from the engine is connected to the casing of the torque converter.
This makes the entire casing rotate at the same RPM as the engine.
The pump is connected to the casing and thus rotate with the same rpm and direction.