iCatcher--$2, from Joeisanerd.com--also offers syncing capabilities between iOS devices, but not to quite the same extent. The app will sync playback times for podcasts so that you can interrupt This American Life on your iPhone and pick it up at the right moment on your iPad. Unlike Downcast, though, you can't sync your subscriptions--you'll have to subscribe to a show on both devices to accomplish the syncing.
Vemedio's $2 Instacast--which is the most video-centric of all the podcasting apps I viewed--also offers syncing between devices, but with a catch: iPad owners have to pay $5 for Instacast HD, the standalone iPad-optimized version of the app. It's a limitation of the App Store that existing users of one version of Instacast can't upgrade to the other, but you can always just install the iPhone version on your iPad: It will still sync with your phone, and if there's any difference in video quality, it wasn't noticeable to me. Nevertheless, this is a hurdle you won't have to jump have to jump through with either Downcast or iCatcher.
The rise of AirPlay streaming makes it easy to play music from your iPhone or iPad wirelessly to a great stereo system or an Apple TV. That can also be done with podcasts--though not every third-party app opts into Apple's requirement that they be optimized for iOS 4.3 or later to support AirPlay streaming.
Downcast, iCatcher, and Instacast all stay in the running under this criteria, but several other good apps join the race here. I was a big fan of Alex Sokirynsky's $2 Podcaster app when I reviewed it a year ago; it has remained my player of choice on my iPhone. It lacks the iPad optimization and syncing, unfortunately, but its AirPlay compatibility keeps it useful.
The same goes for Pocket Casts, a $2 offering from ShiftyJelly. The chief attraction of this offering is its elegant-looking user interface and its AirPlay compatibility. But it is built mainly for the iPhone, just like Podcaster, and I don't need two apps with the same capabilities and same limitations on my iPhone. More for sentimental loyalty than anything else, I stick with Podcaster--but either will do the job just fine if you only listen to and manage podcasts from your iPhone or iPod touch.