The Export / Import route will export and import your site’s content — all pages, posts, any custom posts, images (if you tell it to grab them during the import phase), and any associated metadata for the various posts & pages. (Note that you’ll need to make sure any custom post types are set up on the new site before you run the import.)
The MySQL dump and import will get everything for your entire site, including which plugins are enabled*, what theme is in use*, all the settings, all the users, etc.
References
Migration
Also, if you haven’t already Googled wordpress migration best practice, it turns up a number of suggestions:
- https://codex.wordpress.org/Moving_WordPress
- http://www.wpexplorer.com/migrating-wordpress-website/
- https://www.sitepoint.com/how-to-migrate-a-wordpress-site-to-a-new-domain-and-hosting/
- https://premium.wpmudev.org/blog/moving-wordpress-site/
…were all on the first page of results for me.
* Note that the database export won’t install the plugins and themes for you; you’d need to make sure you copy the relevant directories and files to the new server.