The website is the original source and the more important entity, so the goal is to bring people there. The blog that we manage is on a larger site called patheos.com, a religious website.
I'm not 100% sure if it's creating a "duplicate" content problem but I am feeling like there might be a uniqueness problem.
Both pages (the website and blog) exist in order to help promote the podcast with the blog posts linking back to their respective full transcript posts on the website.
So I'm thinking the other issue might be that the content on the blog if not duplicate, then is considered "thin". It is wordpress based and the content it includes is made up of posts, and there is one for each of the video and audio versions of the cast. The video version includes the video and and then a few short paragraphs talking about the topic at hand being discussed in the podcast. And the audio version is just one paragraph or so about the topic along with the audio. Technically unique from the video, but obviously short, and is generally targeting the same thing.
The website is also wordpress based and has a post for each of the video and audio versions of the cast as well. The video post just has the video and then the verbatim transcript, like Moz's whiteboard fridays! And then the audio version includes a short paragraph or so on the topic, again technically different or unique from the video transcript and also different from the other audio post on the blog but also "thin". Sorry if this is confusing... 
Thanks so much for your responses so far, I greatly appreciate it!