The purpose of the_author is based off the old functionality of "get_the_author()". It is not deprecated, but is developed to echo the value from get_the_author and as a result of any old theme that might still use the old behavior will also pass the value from get_the_author().
The purpose of author_link() is to display either the authors link or the authors name. If the author has a home page set than you will be able to echo an HMTL link or otherwise, just echo the authors name.
I would say these are similar as they both produce the same outcome with the name of the author although author_link() can be used to provide the URL of the authors website. When processing these functions is when the differences come in as it seems that author() has the use of parameters by passing in values such as strings and boolean. The process for author() has a little bit more work to it rather than the_author_link() as you echo it in a PHP file, you then will be able to display it on your page, inclosed by php tags.
Overall, these are both very useful tools when dealing with blogs, considering that the author is extremely important with blogging websites. Those credited may want more of a following so being able to display the name can be helpful to both the website and the person witting the blogs. Id say getting the author_link() would be the much more easier approach to display the name of the creator.