Use cases for “online” authenticated encryption?
The usefulness of online AE (locally):
Assume you wrote a program that encrypt arbitrary files. Now further assume the user wants to view a movie, encrypted with this tool. The tool can now use the online-property to stream the movie in real-time as it uses online-encryption.

The usefulness of online AE (programatically):
Assume you want to process sensitive data. If your cipher supports online AE it can wipe the data after having written / read it an thus presents less attack surface.

The usefulness of online AE (network related):
Assume you stream a movie. You can now use the online AE feature to minimize the RAM needed (and thus give more RAM to the video decoder) and you can present the user with better loading times.

All in all:
Online-AE reduces memory usage and latency for applictions.