Drupal Form API provides a standardized method to show, validate and submit a form in your Drupal site. It allows other modules to interact with the form you design and alter or add fields to modify the form behavior, if that is needed.
By using the form API, you get to output forms that are fully wrapped in classes and ids that your theme can style in a way that fits your site's graphic design.
The Form API outputs form that are secure, meaning that it makes sure that the form the user submits is generated by your site and is not part of a bogus page designed to feed malicious data into your database. The validation process allows a programmer to sanitize any data to avoid SQL injection and other potential threats to security. It also provides a way to avoid that multiple user modify the same content at the same time.
I also think that building a form as a structured array makes the form design and maintenance more straightforward.
These are only my personal highlights about the use of form API. You might like to read more at this link in the Drupal Form API documentation.