Liquidated damages are what you pay when you cannot satisfy a contract. The amount is usually written into the contract. By paying the amount stated, you are then free of any further action against you for failure to complete the contract.
Penalty implies not that you cannot complete, but that you were slow or your work left something to be desired. You still must complete the job satisfactorily, so you aren't out of the woods yet.
An example would be you are under contract to send me 1,000 shirts by the end of January. If you let me know you can't until February 15, you have violated the contract. Liquidated damages means you would then pay me the agreed upon sum and we are done - you owe me no shirts, nor can I get any more money from you.
A penalty would mean you pay me for the shirts being late, but I will still accept them and you must still send them. So even though you are paying a penalty, we still have a contract for the shirts, they are just going to be late.
The website posted below gives a definition of liquidated damages.