http_response_code is basically a shorthand way of writing a http status header, with the added bonus that PHP will work out a suitable Reason Phrase to provide by matching your response code to one of the values in an enumeration it maintains within php-src/main/http_status_codes.h. Note that this means your response code must match a response code that PHP knows about. You can't create your own response codes using this method, however you can using the header method.
In summary - The differences between "http_response_code" and "header" for setting response codes:
1. Using http_response_code will cause PHP to match and apply a Reason Phrase from a list of Reason Phrases that are hard-coded into the PHP source code.
2. Because of point 1 above, if you use http_response_code you must set a code that PHP knows about. You can't set your own custom code, however you can set a custom code (and Reason Phrase) if you use the header method.