If you have a podcast, would you use a Facebook group or would you use a Facebook page?
Whether you have a Facebook page or group really depends on what you want to accomplish with it.
Facebook pages are great for creating an official presence of your podcast on Facebook. There's even a “podcast” page type now! It's easy to post to pages and it's a good way to continue sharing your content. Facebook pages can also have special features, such as built-in opt-in forms, customization, and live video (someday).
Facebook groups are great for community. It's very easy for members to start and participate in conversations in a group. Plus, group posts and notifications are more prominent than those from pages. But groups don't have the special features that a page can have.
Thus, I recommend you make a page for your podcast, and consider a group if you want to create a community and don't mind it being on a platform you don't own (Facebook).