Streams are the more important one. Once you embrace streams in your applications a lot of things become easier. For example here's a small script I wrote at work to process some csvs and do stuff with them
https://gist.github.com/jb55/0ea6aba86e269f1e526b
Needless to say before I knew about streams that program was twice as large and twice as buggy. Also check out substack's stream handbook to learn more, they are really handy.
If you are doing something less "frameworked". Something like custom (text/audio/video) (processing/compression/analysis). You would probably be reinventing Streams if you're not explicitly using them. Meanwhile, apart from implementing Streams, I can't think of a good usecase for EventEmmiter that isn't already in a framework like Express or Socket.io (Someone please enlighten me).