Q: When I download a song on my computer why does it not automatically sync to my iphone?

You have to have Automatic Download enabled.

On the iPhone go to Settings > iTunes & App Stores > Enable Automatic Downloads for Music, Apps, and iBooks, also you can make the choice if it should download over cellular data.

iTunes: Edit> Prefrences > Store > Enable Automatic Downloads.

This is Apple's article on it.

http://support.apple.com/kb/HT4539