Admin Tools never interferes in any way when you are updating your site. Don't forget that Joomla! Update, the component tasked with updating your site included in Joomla! 2.5 onwards, is actually the Joomla! updater Admin Tools used to have, something which we contributed to the Joomla! project. If you have troubles using Joomla! Update make sure that you have not manually removed the .htaccess Maker exception for the administrator/components/com_joomlaupdate/restore.php file we have put there by default. If you did, what you're experiencing is a self-inflicted problem.
As for moving your site to a different host, yes, there will be problems under two circumstances:
1. You are using the .htaccess Maker to generate a .htaccess file. Due to the way Apache (your webserver) works we have to use the hostname of your site. This is documented as the first thing to check in https://www.akeebabackup.com/documentation/troubleshooter/prbasicts.html
2. If you are using the administrator password protection. Again, due to the different ways Apache works across different platforms / servers the generated .htpasswd file may not be readable in the new server. This is documented in https://www.akeebabackup.com/documentation/admin-tools/admin-pw-protection.html
Both of these issues come from the way your web server (Apache) works. A solution is not up to us. All we can do is document the potential pitfall, which we have already done.
We'd appreciate it if you did ask for our support before filing a review. We could have explained as much and told you exactly what you need to do (or point you to the documentation pages with more thorough instructions).
Thank you for the excellent extension AEC.