You'll need to be a Rhapsody subscriber to take advantage, but that's true of the streaming service too. Luckily for you, Rhapsody just dropped its monthly fee from $15 to $10. That's for one mobile device — you'll need a more expensive subscription if you want support for more.
While Rhapsody's app is the first in the App Store to let you download your music like this in the U.S., Europe-only service Spotify actually released this feature first. It just isn't available in the U.S. yet.
The downloadable music (and the app in general) will become much more useful when Apple adds multitasking to the iPhone and iPod touch this summer. Then you'll be able to keep listening to your Rhapsody playlists even while you're using other apps. Rhapsody also has plans to make individual tracks and albums downloadable, not just playlists.
One final note: The iPhone and iPod touch app works on the iPad as well, but Rhapsody just announced that it's going to launch a special version optimized for the iPad. That will happen this fall when Apple adds multitasking to the iPad's operating system.