Difference between new page, new post and new showcase

Pages, Posts and Showcases are just examples of different Contenttypes, and all of them can be replaced with whatever you like.

is an example to emulate what you might need / need for standard site pages.

Writes an example of something like a blog post.

Showcases are an example of a useless Contenttype that provides an example of how you can use most of the available field types.