Originally a money-box in the shape of a pig, the term "piggy bank" has become a lot more general, to refer to any object intended to hold money and with a slot for inserting the money. It is a term used by children, or perhaps by adults to children. It is also used by adults in a frivolous mood, usually figuratively.
In contrast, a "money box" may be simply a box that is intended to contain money, or any box used to contain money.
Just this morning I paid for my breakfast using only coins--nearly $5.00 in coins. I said to the cashier after handing her the coins, "There, you've got my piggy bank."
A piggy bank would normally only hold coins.
A "money box" or more usually a "cash box" would have both coins and paper money.
A piggy bank is a child's bank and my quote was an effort to be amusing (a failed effort, I might add). You would not expect an adult to have a piggy bank; and you would not expect a child to have a money box or cash box.