If you have updated your iPhone backup with iTunes or iCloud, you are able to move most of the important data from old iPhone to new iPhone by restoring new iPhone with the backups.
When the restoring process is finished, the data will be imported into your new iPhone, but the media files, such as music, videos and so on, won't be transferred. If you have purchased albums in iTunes Store, you will have the songs in your purchased history, and you are able to download the songs on your new iPhone directly by entering iTunes Store with your Apple ID. You'll need good Wi-Fi connection to finish the task.
These methods are helpful for transferring data from old iPhone to new iPhone, but they won't help you transfer to the non-purchased files, like CD copies, ripped videos and so on. At this moment, you will need the iPhone transfer software to transfer non-purchased files from old iPhone to new iPhone.