Ask a 100 people, and you will get 100 different responses (although there aren't 100 different apps out there). 🙂 However, I have tried many of the apps out there and I have settled on Instacast as my primary. I enjoy Instacasts clean interface, downloading and refresh speed (it can download multiple podcasts at once), push notifications of new episodes (many others have this as well), and iPhone and iPad versions of the app (even though separate expense). Coming in a close second is another favorite by most, Downcast. I find Downcast a little laggy in scrolling and its refresh doesn't seem as quick as Instacast. I also find the playback screen a little too busy and the UI a little too iOS circa 2008-2010.
Pocketcasts is a fine app, but no smart playlists, and no auto downloading of episodes is a show stopper for me. Although speaking to the dev of the app, he has assured that these features (and more) will be in an upcoming v4.0 of Pocketcasts.
Don't even get me started on the total screw-up that Apple's podcast app has been to date. 🙂