These two parts are pin to pin replacement. and the source code are compatible as well.
You can use the same firmware, I have confirmed that you can use the same LPC1837 firmware with the LPC4337 MCU, I tested using the periph_blinky example from LPCOpen for the LPC1837 and flashed it to the LPC4337 and it worked without any problem. If your application created using LPCOpen, then the migration should be fast and the new features of the LPC4337 can be implemented easily.