You need to change a setting on your iPhone that specifies where newly created contacts are placed.
The iPhone has multiple "accounts" where contacts can be added (the iPhone also refers to them as "groups"). There is a default contacts account/group called "On My iPhone". Now that you have added a Google Apps account, you have a second option, because Google Apps has the ability to emulate an Exchange backend and can store contacts.
Once you set your Google Apps account as the default contacts account, then newly created contacts will be placed there instead of in the "On My iPhone" account, and the sync will work as expected.
To change the default, go to: "Settings"->"Mail, Contacts, Calendars" then scroll down the "Contacts" subsection. Click on "Default Account" and change it to your Google Apps account.