@Jeffro -
Ok, that makes sense. But, my question is, what’s the difference between selecting the Quote icon to see all posts with the Quote format and the Quote category listing all posts with that Category? I am assuming that if both exist, that a Quote formatted post would also reside in a Quote category.

At its most fundamental, "Post Format" is simply a taxonomy - just like "Category". The difference is that the Category content defines the description or subject of the content, and Post Format defines the format/medium of the content.

You can certainly use the two taxonomies interchangeably (and likewise for the post-tag taxonomy), and there would be nothing wrong with that. But you gain an extra degree of taxonomization by using the two taxonomies for their different intended purposes.

For example, if you focus on quotes, then you could have a "quotes" category, and do everything with that content that you can currently do with the quote post-format (though you would have to roll your own formatting implementation).

But say you have a "BBQ" category, that could include standard posts, quotes, videos, photos/galleries, etc. It would then be logical to use the post format taxonomy in combination with the "BBQ" category.

Another benefit of the post format taxonomy is that the terms are fixed. Theme/Plugin developers can do interesting things with the taxonomy that would otherwise be more difficult when using arbitrary category terms. For example: creating a widget that displays recent posts with the status post format (or with the quote post format) - or a custom page template that displays images or videos or galleries -is dead simple. Trying to replicate that using arbitrary category terms requires user intervention, such as selecting the category to display.