Audiobookshelf (which is a great project that I use personally for audio books) is focused very heavily on audio books. With features like libraries and all that. I don't feel like it has every feature I'd want for managing podcasts. Especially when it comes to searching and syncing feeds to things like antennapod. This project focuses on being the best podcast managment server it can be. It makes browsing podcasts, feeds, and episodes a breeze. Adding podcast feeds is a search away and that's all integrated right into the app. This app is also built in rust rather than javascript. Which some people really like.
Audiobookshelf is a phenomenonal piece of open source software and if it does everything you need it to I think that's a great solution! If not, this software might be a good fit.
Pinepods also supports podcast 2.0 podcast specifications and wiol be implenting more of it such as chapter support and timestamp support very soon. I don't think Audiobookshelf does? Though I'm happy to be wrong about that as I don't use Audiobookshelf for podcasts.