A street address is significantly more specific than a zip code. If a match is solely based on a zip code, it's just a point at the centroid of or randomly placed within the zip boundary. Your question doesn't specify if you are referring to address and zip code as elements of a locator, a particular locator (or style), or parts of a composite locator.
Matches are going to depend on the format and quality of the data. If you have street addresses with no zip, they can be matched by address but not by zip. The reverse is also true. It's important to understand that (depending on the locator being used) a match isn't determined by any one part of the address, but the combination of available parts that most closely matches the locator records.
I would suggest reviewing the ArcGIS help files on Geocoding, and specifically the one on process.