Agree with the above, they're not exactly equivalent. One is a reading software, they other is a distribution service.
You need to produce EPUB for either.
Adobe Digital Editions is an ebook reader that supports ebooks in PDF and EPUB formats.
Overdrive is a distributor (mainly libraries) that provides ebooks in many formats, including EPUB.
Ebooks can be with or without DRM. There multiple DRM technologies out there.
Adobe Digital Editions only supports DRM'd EPUBs and PDFs that use Adobe Content Server DRM (aka ACS).
Overdrive distributes EPUBs (and PDFs) with and without DRM. For those that require DRM, Overdrive also uses ACS.
Therefore, a DRM EPUB that distributed through OverDrive would be supported in Adobe Digital Editions.
Maybe you're referring to something else...but it's not clear based on your OP.
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