William's answer is correct. We use the same codepath with Copy_management as we do with right-click 'Copy' - right click 'Paste' (or control+C, control+V). This maintains things like domains/relationships etc, it is actually making a copy of the data and its other related data and pasting it. Feature class to feature class is essentially an export of the data. This only copies what you specify in the tool. Copy preserves geodatabase maintained values such as OID or GlobalID while exporting data will not maintain these values.