From what I can find, it appears that Media Type is going to be the new terminology for MIME Type in the future.
This according to this page https://www.iana.org/assignments/media-types/media-types.xhtml which says, "[RFC2046] specifies that Media Types (formerly known as MIME types) and Media Subtypes will be assigned and listed by the IANA."
So it appears that they are, for the most part, one in the same. As for how Tika implements them, I'd suggest going over the documentation. Here are links to the pages for MediaType and MimeType from Apache Tika 1.17:
From personal experience, the Tika MimeType is very trustworthy for file extension verification, so I'd say to keep using that for now, but keep an eye out from Apache to make sure it doesn't get deprecated/become obsolete as MediaType becomes the main terminology.