It can take some time for all of the data to be updated after the sync is complete. Easy Phone Sync updates the contacts first, then the messages and finally the calendar. This process starts after the Galaxy device is disconnected from the PC or Mac.
You can check if Easy Phone Sync is processing messages, contacts, etc. by pulling down the notification bar on the Galaxy device (the thin bar at the top of the screen displaying the time and battery status). A progress message will be displayed if Easy Phone Sync is active.
Note that Easy Phone Sync cannot transfer content backed up to iCloud. If you have been backing up to iCloud it is necessary to create an iPhone backup on your PC or Mac in order to transfer the data using Easy Phone Sync.
If none of the above apply then restart your phone. This will force Easy Phone Sync to start processing any copied data.