It often fails to download podcasts. Overcast I can just tap 'download' and trust that it will do its work. Not Apple Podcasts.
I got the "this episode is not currently available" message all the time whereas I never get anything similar from Overcast.
It often fails to transition from one podcast to the next in the playlist (ie, stops playing). Sometimes even when all involved episodes have been downloaded.
It fails to automatically delete old episodes.
It fails in multiple ways when it loses connectivity. E.g., it refuses to continue playing a very well-buffered episode if there's no connectivity. Mind you, I suspect that some of the connectivity issues may be a result of them never having tested it with 'slow' connections (2Mbps in my case). This obviously wouldn't absolve them in the least, however.
This is just off the top of my head. It's one of the worst pieces of software I have ever used. From a company with billions in cash.