First major missing functionality I noticed (relative to Downcast) — the list of podcasts doesn’t offer any clues as to which ones have unplayed episodes. You have to click each one to see whether there are new episodes available.
[edit: evidently I wrote this too soon after my first use of the app. Once you have finished an episode of a podcast, it appears in a new section further down the main list, titled "Played Podcasts". More on this below.]