Take Execute Mode for example. In 10 years of Vim use, I've never tried it. Let's fix that now.
You enter Execute Mode by pressing capital Q (shift + q) when in normal mode. You'll see this at the bottom of your window.
Entering Ex mode. Type "visual" to go to Normal mode.
This seems familiar enough, but this is different than when you simply type ":" to enter a command at the bottom of the screen in normal mode. In normal mode, typing ":10 " + enter takes you to the tenth line in the document. In Execute Mode, ":10" + enter displays the tenth line just below your cursor. Hitting return again shows the eleventh line, and so on. You are not editing the document in this mode, simply accessing it. Interesting, but not yet terribly useful. Let's see what else you can do.
This writes the file, just as it would if you had typed :w filename in normal mode, except again, instead of returning you to editing the document, you remain in Execute mode, meaning that the ":" appears again, awaiting your next command.
I'm afraid I can't imagine too many good reasons to use Execute Mode. The best use-case I can find is that you could use the :n, where n is a line number, feature view a section of the document you are not working on without leaving the section you are working on. If you have any tips on how Execute Mode makes your life easier, please leave a comment.