Mikhail Madnani at Beautiful Pixels has a good rundown of the two best (IMO) podcast apps on iOS.

Instacast 4 for iOS builds upon its earlier wireframe and brings an elegant coat of paint to fit right at home on iOS 7. The animations have a lively bounce to them and the app UI is simple. Pocket Casts 4, on the other hand, has been rebuilt from the ground up for iOS 7 and if you’ve used the Android app, you will feel right at home with this one.

While I think Pocket Casts is fantastic, I’m still sticking with Instacast because I use the Mac app a lot. It’s got its quirks—which are infuriating at times—but I like the UI more than anything else available.