The Map module may not work in case the allowed referrers are defined incorrectly. The following error message will appear in the browser console

To check allowed referrers open Google Developers Console. Open a project that you own and then click on the Credentials button and on your API key:

In the settings you will find the Application restriction option where you can add your domain name:

Make sure that protocol is the same as on your website, HTTPS or HTTP. You can also add the domain name without protocol, like this:

*example.com
*example.com/*

For testing purposes you can set Application Restrictions to "none". 

Note: Changes may not appear instantly and there could be a delay before Google updates the data. Usually it takes a few minutes.