Here are a few differences off the top of my head.
MDF:
Cheaper to purchase
Heavier
Smoother surface which can be useful for painting or veneering
Edges are prone to damage
Prone to water damage unless it's a waterproof variety like Extira
Poor screw holding ability on the edges
Plywood
Holds screws better on the face but not much better on the edges
Less prone to sagging
More resistant to flexing and less likely to break under flex load
MDF:
Cheaper to purchase
Heavier
Smoother surface which can be useful for painting or veneering
Edges are prone to damage
Prone to water damage unless it's a waterproof variety like Extira
Poor screw holding ability on the edges
Plywood
Holds screws better on the face but not much better on the edges
Less prone to sagging
More resistant to flexing and less likely to break under flex load