Let me preface with what each of these are:
Drush is a command line shell and Unix scripting interface for Drupal. Drush core ships with lots of useful commands and generators. Similarly, it runs update.php, executes SQL queries, runs content migrations, and misc utilities like cron or cache rebuild.
Composer is a tool for dependency management in PHP. It allows you to declare the libraries your project depends on and it will manage (install/update) them for you.
You cannot use Drush to update/upgrade Drupal 9. That particular Drush functionally/feature got deprecated. As Drupal 9 requires Drush 10 or higher, it doesn't have it.
source: Updating Drupal core via Drush
Composer is a very popular PHP tool used outside the Drupal world to manage dependencies.
Drupal decided to adopt it. Once this happened, there was no need for Drush to re-invent the wheel when there is already a dedicated tool built for this job specifically and that does it very well.