Fantastico is great for a fresh install. After that, it has a reputation of being unreliable.
In the /docs folder are several resources to explain things like installation, upgrading, and other things you'll find useful.
Your upgrade has only touched some of your files. It has not touched any of your customized files, meaning that they won't have any of the new features/fixes in them that might have been created during the 1.3.7 release. As such, you need to follow the upgrade instructions in the /docs folder of your Zen Cart folder.
You should also download the latest version of Zen Cart manually. See our home page for the link. You'll need the raw files in order to understand some things, and also in order to do upgrades.
Your upgrade has not handled the database side yet, which is why you're seeing the warning.
You need to install the zc_install folder from the Zen Cart distribution files (ie: the zip you'll download from our home page). Once you've uploaded it via FTP, you can click through it, choosing "Database Upgrade" instead of "Install" on the System Inspection page. Then just follow instructions.
Then go back and review your customizations, using the upgrade instructions from the /docs folder. You'll need to WinMerge the updates into your customized files or else you may end up with some odd behaviour on your site.