EFS and S3 have slightly different use cases.
S3 is meant as storage for an application. An application can read and write from S3 using a certain API. This API is what is called a REST based API over HTTP.
EFS seems to be meant to be used exactly as a normal filesystem on Linux or Windows (or whatever), and individual computers will attach to it (I assume) basically the same way you normally attach to a shared folder on the network.
It should be mentioned that several developers have provided applications that aims to turn S3 into a sort of filesystem, but none of those work very well. Enter EFS.