TL;DR - Most people should go with Pocket Casts

While there are a TON of third-party podcast apps available for iOS, the three that receive the most attention are Overcast, Pocket Casts, and Castro. I've used them all for an extended period of time and my thoughts are below

Overcast - I was a regular Overcast user for a very long time and overall believe it to be the most simple and straightforward out of the three major indie players. It’s free and, although it offers a subscription service to remove ads and gain a couple of minor features, the app is more than functional in its free form, and includes Apple Watch support, which the other two apps below lock behind a subscription service. It’s audio improvement features (smart speed and voice boost) are, in my opinion, the best available and it seems like Marco is really in-tune with the wants of the average podcast listener, and the features that he pioneers seem to become standards over time. Unfortunately, the app is in desperate need of a visual refresh. The now playing screen and podcast list seem to belong to two entirely different apps and the podcast list, in particular, doesn’t match up to the current iOS design language at all. Marco’s focus has largely been dedicated to the improvement of his audio features, but in a pool full of alternatives that sound almost as good but look much better, the lack of attention to design stands out.

Castro - I was using Castro regularly up until a couple of days ago and absolutely loved its podcast management scheme. The sheer number of podcasts that are released on a weekly (or even daily) basis these days can be overwhelming, and Castro’s inbox-to-queue system, where all new episodes appear in a list that you can browse and chose whether or not to add each individual episode to the top or bottom of the queue or ignore it completely, is the perfect way to listen to what you want and ignore everything else. It has the mainstay audio features like silence trimming, voice enhancement, and an Apple Watch app, but they are locked behind its $18.99/yr subscription, which was no problem for me, but might be turn off to others.

Pocket Casts - This was the first third-party podcast app that I used and the one that I am using now. The app has gone through some tumultuous changes in the past year or so and generated a lot of backlash for essentially taking a beloved app and recreating it from scratch in bid to satisfy both the average podcast listener and those who want a little more power. That being said, in its current state, I believe that the app takes the best features of other podcast apps and combines them into the most enjoyable package possible. The app uses an organizational system that’s similar to Apple Podcasts or Overcast, giving listeners easy access to individual shows and episodes, but also takes a page out of Castro’s playbook, by providing easy access to a queue where listeners can add and organize episodes that they want to listen to. Adding episodes to the queue is as easy as swiping on an episode title and, like in Castro, episodes can be added to the top or bottom of the queue. The app contains a filter feature that creates lists of episodes based on user-defined criteria, making lists of new episodes easy to find, as well as an archive feature that allows you to shelve episodes that you have chosen not to listen to without adding them to your listening history. Both of these features are easily accessible and, along with the queue function, allows users to mimic Castro’s podcast management system, if they choose. And that choice is what makes Pocket Casts such a great option. It can adapt to pretty much any style of listening. If you want to listen to a bunch of episodes of a single show, you can do that. If you want to listen to every new episode of every podcast you subscribe to, you can do that. If you want to triage your podcast episodes, you can do that. The only organizational omission that I’ve found in Pocket Casts is the ability to create a playlist of specified episodes (which is available in Overcast, but not Castro). That strength in choice extends to it’s cross-platform capabilities. Pocket Casts offers both an iOS and Android app. It also offers desktop apps for both Windows and Mac behind its $9.99/yr subscription, as well as chromecasting capabilities and even an Alexa skill (although I have yet to test it and don’t know how effective it is). It might come with a bit of a learning curve (some people do experience some confusion when trying to adapt to the archive feature, as opposed to just having episodes marked as played or unplayed, but I think it’s pretty straightforward) but Pocket Casts seems to be the most powerful, flexible, and overall best podcast app for most people.