The function noted above is designed to do exactly what it says: grab the author of the post/article. This function, as stated in the WordPress Reference, "is not deprecated, but is designed to echo the value from get_the_author() and as an result of any old theme that might still use the old behavior will also pass the value from get_the_author()."
This function, like the one above, also acts as a way to display the name of the author. It also can display the author link; meaning that it can be clicked and lead the user to other articles written by said author. This works only if the author's url exists in the user meta, otherwise it results in the same result as the_author().
These two functions have very similar use cases. However, as noted above, the_author_link() serves a more useful purpose in cases where the author is presented many times on the website and has their own url. Otherwise, the simple the_author() function will suffice.
These two functions, while almost identical in use, are provided for the slim difference in cases where the author may or may not have their own url present in the user meta. That once difference can decide which function makes more sense.