Explain the_author() Function
The the_author() function displays the name of the post's author. Authors are set up in the Settings > Users area. The name used is the name
that was set in the user's profile. The syntax for this is: <?php the_author(); ?>
Explain the_author_link() Function
The the_author_link() function shows the link to the post author's website. The site for an author is set in the Settings > Users area
for each author. If there is no link set for the author, the author's name is displayed instead (unlinked). The syntax for this is:
<?php the_author_link(); ?>
Explain Differences and/or Similarities
The functions above look the same, and at times will provide the same output but are for different purposes. the_author() is only meant to display the post author's name. the_author_link() is intended to display the post author's website link. But as stated above, it will display the author's name if no link is available. There are two main similarities in these functions. The first is that both must be used inside The Loop. The second is that both of them are found in wp-includes/author-template.php file within WordPress.
Summary of the Documentation
These two functions are useful to display author meta data where desired when you are displaying a post. It can be used on a posts list page or a single post page. Depending on the situation, you can decide whether to display the author's name only or include a link to their site.