the_author() Function
This displays the author of the post. It is a legacy way to accomplish this. The current way to do this is get_the_author(). The the_author function will still call the same info as get_the_author if I am reading the developer documentation in WordPress.
the_author_link() Function
This function displays the author’s name as a link to their homepage if they have one. If not it just displays the author’s name. This is great if the author maybe has a site outside of the site to which they are posting and the site administrator allows them to promote themselves.
Similarities and Differences
Both of these functions display the name of the author, and it is in the display name field in WordPress. You would use the_author() if you do not want authors to self-promote on your site. Use the_author_link() if you allow authors to self-promote or if you want to direct visiters to the author’s home page.
Summary
Both are ways to display the name of the authors of posts on a WordPress site. If you are a business who wants the visitors to stay on your site use the_author.