iOS's Music app, much like iTunes on computers, has traditionally been the destination for all audio-related content - here you'd find the music synced from your iTunes library, podcasts, and audiobooks.
And that's very much the theme of iOS 6's Music app - from all appearances it's lost more than it's gained. The single new element is a grayer interface with the iPhone and iPod touch version. In the Lost column are any references to Apple's now-defunct music social-networking service, Ping; the capability to download single iCloud Match tracks; and, again, podcasts and iTunes U content appears to be missing.
"Appears" is the key word here. As mentioned, if you've installed the Podcasts and iTunes U apps, that's where you'll find this content. However, if you haven't installed these apps and you sync, within iTunes to your device, podcasts and iTunes U content, you play that content within the Music app. Just tap More and you'll find entries for both kinds of media. (You'll also find an Audiobooks entry here if you've synced audiobooks to your device.)
This is fine as long as you sync this content from iTunes. However, if you were hoping to obtain podcasts or iTunes U content from the iTunes Store app and play it in the Music app, you'll find that under iOS 6 you're out of luck--neither kind of media is downloadable from the iTunes Store app. To directly download this material from Apple you must install the Podcasts and iTunes U apps and use them to obtain the content you want.
Other than the aforementioned emphasis on gray and silver hues, the functionality of the Music app hasn't changed. This is still the app you use to play music stored on your device, as well as to stream and download iTunes Match tracks. Cosmetically, a few icons have been rearranged: On the Now Playing screen, for example, the Volume slider and AirPlay button sit next to each other; on an album screen the Download All button has been moved up next to the album art; the Shuffle button bears only the Shuffle icon rather than the word "Shuffle" as it did previously; and the icons throughout are generally smaller.
More notable is the absence of buttons to download single iCloud Match tracks. You can still download all the tracks from a playlist, an album, or a particular artist, but the capability to grab individual tracks is gone. (You can download individual songs you've purchased from the iTunes Store via the iTunes app, however.) This could prove inconvenient for those who prefer to download tracks on-the-go via a cellular connection, as you're now compelled to download an album's worth of tracks when you may want only one.
Those few people who used Ping may be disappointed to find that on their iPad they can no longer recommend a track or create a post based on a currently playing track. (If you weren't aware of the feature, under iOS 5 you'd find the Thumbs Up and Post icons on either side of the timeline on an iPad's Now Playing screen.) Given that Apple has officially deep-sixed Ping, those icons would do you very little good now.