I am using using a WordPress plugin that generates a tarballed .sql file
as part of a database backup routine.

I am also using a Drupal module that generates a .mysql file on a different site as part of a database backup routine.

Any difference between the two file types?
Does PHPMyAdmin import both files the same way?

Look at the content.
99% chances there are the same: scripted SQL commands to restore datas and structures.

The file extension is meaningful only to windows. Every other systems I ever worked with don't use nor need a file extension to recognize the content of the file.
So, I would not base myself only on an extension to recognize a file source.