Correct, unzip the downloaded backup. Inside is a "Database" folder. Inside that is your database backup in .SQL format. Just import that using phpMyAdmin.
If the URL is different you need to follow: http://codex.wordpress.org/Moving_WordPress
Editing wp-config.php is not enough as some areas of the database actually point to the old URL, and you'll find the WordPress login just doesn't work and redirects you to the old URL.
I think on the one or two I've done myself, I just needed to edit wp_options table for: home and siteurl options. Once done, I then needed to deactivate a few plugins and reactivate them as some plugins were also storing various URLs. After that everything seemed to be fine, but it depends on the blog, plugins installed, themes installed, etc etc. It's not really my area of expert at the moment!