FaaSes do add some twists on CGI. In CGI the short-livedness is the process, in a FaaS it's the container. A CGI program can get away with relying on local modifications to a file system in a shared environment. FaaSes somewhat point away from that.
There's also the speed of scaling. Insofar as FaaSes rely on lower layers to insulate the layer between process and machine, it becomes plausible to grab a number of machines relatively quickly. There's a cold start cost, to be sure, but the cold start on "launch this container" is probably better than "install this software and then launch it".