Moodle web and Moodle apps currently support FontAwesome 4.7.0. However you can add code to support FA > 5.
<link rel="stylesheet" href="https://use.fontawesome.com/releases/v5.2.0/css/brands.css" integrity="sha384-nT8r1Kzllf71iZl81CdFzObMsaLOhqBU1JD2+XoAALbdtWaXDOlWOZTR4v1ktjPE" crossorigin="anonymous">
You can try that to add support, however I'm not sure why is working on your website (problably 4.7 has any compatibility). Please check it again and check the fontawesome documentation for further help.