PSoC Creator can smartly detect if your device is compatible with the project even though it isn't exactly the same chip you have connected. This generally applies to chips which have almost identical designs and belong to a particular series class. This isn't the case with PSoC Programmer because it requires the chip to be exactly the same as the one in the HEX file. PSoC Creator says this before you program as higlighted below if you have a different target other than the one in your project:
This is just a feature of PSoC Creator to help developers to test with smaller flash size MCUs and larger ones belonging to the same family. So it isn't stringent upto the last digit in MPN.
Just build the project for CY8C5666AXI-LP004 in Creator and then program the HEX file using PSoC Programmer. It will work.