I just bought a new computer. I opened my iTunes account and authorized the new computer. However, the library does not include all of the music that is on my phone. Why aren't they the same? And how do I get the music from my phone into my computer's library?
MacBook Pro with Retina display, iOS 8.3
The only things which will automagically download to your computer are items purchased from the Apple Store. To get the remainder you need (and really should have) transfer the library from your old computer. If you no longer have that you will have to buy third party commercial software to get non-iTunes Store items from the phone to the computer.
Your i-device was not designed for unique storage of your media. It is not a backup device and media transfer was planned with you maintaining a master copy of your media on a computer which is itself independently backed up against loss. To use a device with a different setup you transfer the old library from a computer or a backup directly to the new setup, not the device to the library. Media syncing is one way, computer to device, updating the device content to the content on the computer, not updating or restoring content on a computer. The exception is iTunes Store purchases which can be transferred to a computer.
iTunes Store: Transferring purchases from your iOS device or iPod to a computer - http://support.apple.com/kb/HT1848 - only purchases from iTunes Store
For transferring other items from an i-device to a computer you will have to use third party commercial software. See this document by turingtest2: Recovering your iTunes library from your iPod or iOS device - https://discussions.apple.com/docs/DOC-3991
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