The init hook is triggered a little earlier than the admin_init. Between them is triggered wp_loaded and few hooks related to updating the DB.
The init hook fires when WP is finished loading (viewer-facing and administration side).
The admin_init hook fires on administration side only (/wp-admin/*), therefore is triggered also by front- and back-end ajax requests (/wp-admin/admin-ajax.php) and POST handler (/wp-admin/admin-post.php).