Running PHP as a CGI means that you basically tell your web server the location of the PHP executable file, and the server runs that executable
whereas
PHP FastCGI Process Manager (PHP-FPM) is an alternative FastCGI daemon for PHP that allows a website to handle strenuous loads. PHP-FPM maintains pools (workers that can respond to PHP requests) to accomplish this. PHP-FPM is faster than traditional CGI-based methods, such as SUPHP, for multi-user PHP environments
However, there are pros and cons to both and one should choose as per their specific use case.
I found info on this link for fastcgi vs fpm quite helpful in choosing which handler to use in my scenario.