What do the js-text-full and text-full in the StringTextAreaWidget class do?
These are classes that Drupal sets for its own css and js. For javascript it is always with the prefix "js". At first sight this seems to be redundant. The reason is, if someone wants to override the css class, it doesn't effect the js behavior and vice versa.

If you want to address the field through your own css, you can always target the field name "edit-mytextfield". If you use a class with the name of a text format, there is a chance, that you target other fields you don't want.