iOS podcast app that differentiates between and gives different functionality to long/short presses of the play/pause button on the headphones Overcast for iOS has a feature that allows you to either skip forward in the track/podcast or move forward in the playlist of tracks/podcasts depending on wether you just do two short clicks on the headphones play/pause button or one short and one long. For moving backwards in the podcast or in the playlist it's three short clicks or two short clicks and one long click, respectively.

I'm having trouble with the sync/download functionality of the app, so I'd like to switch, but I'd really like to download an app with the features described above. What other podcast clients for iOS, if any, have this feature?
Those are system wide commands, not exclusive to your app:

Fast forward
To speed up your track simply tap the Play/Pause button rapidly twice and hold the control down after the second tap -- the track will skip forward.
Rewind
Similarly, to rewind a song tap the Play/Pause button three times and hold the control down for the track to skip back.
Skip to next track
Bored of the song that's playing? Just tap the Play/Pause button twice to skip to the beginning of the next track.
Skip to previous track
You can also skip to a previous track -- all you need to do is tap the Play/Pause button three times.

http://www.computerworld.com/article/2476507/apple-ios/ios--how-to-use-iphone--ipad-earphones-in-13-tips.html
Some of them even work on your Mac.
The Apple Podcasts app has the same functionality.