Wordpress - Is There a Difference Between Taxonomies and Categories?

Taxonomies, as previously described are a collective noun for the following

• category

• post_tag

• post_format

• link_category

• custom taxonomy

The first four are built-in taxonomies, while custom taxonomies are taxonomies that are manually created by the user with register_taxonomy. Custom Taxonomies can be hierarchical (like the build-in taxonomy category) or not (like post tags)

The categories and tags that you create in the back end under the 'Posts' screen are in actual fact terms of the taxonomies category and post_tag

You cannot create extra terms in post_format. post_format have the build in terms like post_format_video, post_format_gallery etc. See the codex for all post formats included

EDIT

I have updated the Taxonomies page in the codex to include my diagram and the missing post_format taxonomy

Category, Tag, Link Category and Post Formats are built in taxonomies.

From Codex

Basically, a taxonomy is a way to group things together.

The names for the different groupings in a taxonomy are called terms. Using groupings of animals as an example, we might call one group "birds", and another group "fish". "Fish" and "birds" are terms in our taxonomy. As an example from WordPress, a category or tag (see next section) is a term.

Category is one of the built in taxonomy in WordPress.

so to clear things: taxonomy is a way to group things together.

we can create custom taxonomies in WordPress just like builtin ones. for example Carscould be a taxonomy with terms like

Audi
Toyota
Benz
Mazda

there are mainly two types of taxonomies available which are hierarchical and non hierarchical

Category is best example of hierarchical taxonomy and Tag is an example of non hierarchical taxonomy