This is a great app if you don't want to eat up all of your data streaming podcasts. It allows you to download individual episodes of each podcast and listen to them right from the app itself.
The one drawback to this app is its lack of discovery features. To find the podcasts to download, you really have to search by specific name. That means you have to know what you want before you look for it. The app itself does run a little slow, but not too much that it interferes with functionality too much.
What I like best about OneCast is how clean and simple the user interface is. The app is extremely easy to use. If you're looking for a straightforward, no frills approach to downloading and listening to podcasts, then this is a pretty good option.
Stitcher is great if you don't mind using your data for streaming, or if you have a WiFi connection. This app's two greatest strengths are discovery and its play mode, but it does lack the download feature.
If you are constantly on the lookout for new podcasts to listen to, then this is a really great app for that. It will let you browse podcasts by category. As you listen to and rate the podcasts, the app will create something called a "Smart Station." The Smart Station will include podcasts based on what you like.
The play mode for Stitcher is different too. You can create seamless playlists to listen to multiple podcasts back to back.