Mail MIME just provides integration with the PEAR Mail_MIME class to encode and decode Mime Mails. Other modules can then integrate with it, for example HTML Mail to use it to send Mails.
Mime Mail on the other hand, is the same as those two modules combined, it contains both the functions to deal with mime encoded mails and exposes itself to Drupal so that mails from other modules are sent with it.
Neither of these modules actually allow you, as a user, to send mails, they deal with mails sent by other modules. There is a small exception there, namely that Mime Mail comes with Rules integration that allows to send HTML formatted mails.