What's the difference between the "Mime Mail" and "Mail MIME" modules?
There are many more than 2 ways to send mail with Drupal. But, the modules you are showing are not for sending mail at all, they are there only to encode / decode mime content. 

The problem you are facing is common in Drupal land: Developers re-inventing the wheel. You will often bump into modules that do similar tasks.

Mime mail has more support, is much more active, and has more features, (it can decode incoming html mail if I understood correctly).   I would go for this over Mail Mime with HTML Mail Pear class.

As for sending the emails, I can strongly recommend SMTP Authentication module if you don't want to use Drupal's built-in mechanism.

Hope this helps, cheers!