There are a number of advantages to using Docker over configuring a server yourself. For one, there's convenience. Know how many steps it takes to setup a new Wordpress blog using docker? Two!
docker run --name mysql -e MYSQL_ROOT_PASSWORD=some-secret-string -d mariadb
docker run --name myblog --link some-mysql:mysql -p 80:80 -d wordpress
Of course, that's for a pretty simple setup. This will bind the port 80 of your host system to the docker instance running Wordpress, so you can't host anything else on the same server. But if you want, you can run any number of docker containers on a single server by setting up a reverse proxy like nginx or haproxy.
Secondly, you will gain access to the vast library of ready-to-deploy application images at the Docker registry (https://registry.hub.docker.com). The Wordpress and MariaDB images used above are examples of this. Pretty much any web application that has attracted notice now has Docker images ready for you to use.
The third is, in my opinion, neatness. Your server itself will not have a big stack installed, like LAMP: all your server needs to know is how to run docker containers. After that, you can run any number of services in containers, and they are completely separate. Want to run rails? Add a container. Django? Add a container. If you want to run three apps on different versions of Ruby, normally you'd have to do some pretty messy stuff with rbenv or rvm. Now, you can simply package your app in different containers, and you host won't care!
And finally security: if you deploy an app with a vulnerability and hackers gain root, they will only have root inside a single docker container, and can't get at your host system.
I've been looking into Docker myself recently, and was surprised at how powerful it can be. I've taken Wordpress blogs offline and brought new ones up in seconds, using only a single Droplet. I've written about my experiences: http://blog.romkevandermeulen.nl/2015/01/31/dokku-docker-server-power-control/
I hope that helps you out. I'm happy to discuss it further if you have any more questions. Let me know!