The Music app received a mostly cosmetic makeover with iOS 6, but there are a couple of items under the hood that have changed. If you use Apple's Podcasts and iTunes U apps, you'll soon discover that content for those apps are now directed to those apps and no longer available in the Music app. The bigger change has to do with iTunes Match.
Before, when you played a song stored in iCloud via iTunes Match, the song would be downloaded to your device as you listened to it. And next to each track stored in iCloud was a cloud icon, which you could tap to download the track. With iOS 6, tracks stored in the cloud are not downloaded but streamed to your device when played. And the cloud icons next to each track are gone.
For any iPhone, iPad, or iPod Touch user who is constantly up against the capacity of their device, having iTunes Match stream instead of download songs by default is a welcome change. Actually, iTunes Match places a temporary cache of the song on your phone, so you can continue listening to it should you lose your Internet connection. I haven't been using iOS 6 and the Music app long enough to know when these caches will disappear, but I can tell you that they got wiped out when I turned off iTunes Match, while tracks or albums I downloaded remained.
Without the cloud icons next to each track, it's impossible to tell which tracks you have downloaded and which are in the cloud when browsing your library. (You can go to Settings > Music and under iTunes Match, turn off Show All Music to view only music you've downloaded.) It's also harder to download an individual track without the cloud icon next to each track. The only places I've seen the cloud icon is when viewing an artist or album. If you'd like to download all of the albums of a particular artist, you can tap the cloud icon that sits below the list of albums. And to download an album, tap the cloud icon at the top of the tracks when viewing an album.
You can't download an individual track as easily as before, but it is still possible. When viewing an album, you can quickly tap the stop icon next to each track before the first track finishes downloading. You have to be quick, however, because the next available track will start downloading as soon as the first track is finished. A better way is to create a playlist of a single track and then download that playlist by tapping the cloud icon below the track.
For more on Apple's new mobile operating system, check out our complete guide to iOS 6.