Both films center on a character that pushes themselves to the brink of madness in pursuit of artistic greatness and contain a parent that desires to curtail their success, as well as an abusive instructor. These similarities are analyzed by Fernando Andrés in his video essay entitled Hands & Feet. In it, Andrés points out the many thematic, narrative, and cinematographic parallels, as well as the differences, between each film.  

Check it out below:

One interesting thing about this analysis is that even though the films are about essentially the same thing, the filmmakers took very different approaches toward telling their stories. Aronofsky, a seasoned veteran of the medium, relies greatly on the cinematography of Matthew Libatique in Black Swan to express Nina's obsessive desire to become a "perfect" dancer. However, newcomer Chazelle leans heavily on the editing of Tom Cross in Whiplash to communicate Andrew's drive to become a "perfect" drummer.

However, yes, these films are incredibly similar in their narrative components, and they go perfectly together as companion pieces. So, if you're interested in making a film about obsessive artists that drive themselves insane while pushing themselves toward perfection, then it'd be a great idea to study these films.

Source: Fernando Andrés