Winning levels grants you bonus diamonds on top of your normal reward. These bonus diamonds are stored in the Treasure Chest.
Once you have collected at least 200 diamonds, you can either open the chest or keep playing to collect up to 400 diamonds. From that point on, the amount of diamonds you collect by beating a level will be smaller, as a full chest can't hold all of them anymore.

The price for opening the Treasure Chest will not change, no matter how many diamonds you have collected.
Keep in mind that harder levels award more diamonds.