Upgrading from Admin Tools Core to Admin Tools Professional is by no means different than installing the component. You do not have to uninstall the previous version; in fact, you MUST NOT do that. Simply follow the installation instructions to install Admin Tools Professional over the existing Admin Tools Core installation. That's all! All your settings are preserved.
When upgrading from Core to Professional you sometimes have to install the Professional package twice, without uninstalling anything in between. Sometimes Joomla! does not copy some of the files and folders the first time you install it. However, if you install the package again (without uninstalling your existing copy of Admin Tools) Joomla! copies all of the necessary files and performs the upgrade correctly.