There's a new "sleep timer" feature in the settings, which stops the music when timer ends. For example, you can set the timer and play your favorite tunes for falling asleep. A similar feature is available in the built-in Clock app for iOS and it works for any music app.
The new app icon is less recognizable than the old one and it's also asymmetric.
Here's the old icon:
Apparently, Google's music service will add support for podcasts next Monday.