the_author() Function
This function returns the author's display name as a string, or echos it.
the_author_link() Function
This function will echo either the author's link or their name.
Differences/Similarities
the_author() function is designed to echo the value from get_the_author(), but will also pass the value to maintain compatibility with older themes. It echos/returns only the author's name, whereas the_author_link() will echo the HTML of the author's home page (if set), otherwise it will just echo their name.
Summary
the_author() can be used to echo the post author's name, whereas the_author_link() will echo an HTML link to their page, or their name.