Batting zero here.
I decided to test things, so I turned podcast syncing back on in iTunes and selected some podcasts to sync. Did this on both the iPhone and iPad. The podcasts showed up in the Podcast app, with no sign of it in Music.
"Huh?" I thought. Then realized I'd already subscribed to that same podcast in the Podcast app, mike the test kind of null.
Back to iTunes, find some new podcasts, download, sync, and there they are; in the Podcast app. Along with all the other podcasts I subscribe to. Still nothing to be found in Music.
So, assuming you are seeing podcasts in the Music app, I''m back to the original theory. When you install Podcast, it changes the behavior of Music so that podcasts don't show up there.
I'm unwilling to delete the Podcast app and lose all my current subscriptions in order to find out; so unless you're feeling more adventurous I guess the guess will have to stand.