I've created a Ubuntu NodeJS 8.10.0 on 18.04 droplet.
I've deployed a Node.js app, tested on my computer.
I can reach my app from inside the Droplet.
I can't reach my app from ...
I would bet that your application is listening on 127.0.0.1, as opposed to the public interface. If you run netstat -tulpn you should see the applications listening, and it might look something like this: