First off, you should have Full Keyboard Access enabled for All Controls in System Preferences > Keyboard > Keyboard Shortcuts. Now then…

As a former Windows user, this dialog boxed perplexed me. I see two buttons that appear to be focused. Mind you, no mouse hover state is affecting these buttons. Restart appears to be focused, and Shut Down looks special. If I hit Enter, I would expect that Restart would be hit, but instead Shut Down is triggered.

My problem was that I was using Enter instead of Space. Hitting Enter triggers the default (special) button. Hitting Space triggers the focused button.

Now when faced with this dialog from the Twitter app, I can appropriately trigger the button I want without hitting Tab.

You can also use space to toggle checkboxes on forms. Hitting enter will submit the form.