I created Drone because I wanted a Continuous Integration system that empowers developers. I've worked at a few large companies, and at nearly every large company the developers have little or no control over the build environment. Every time we needed to install software (node, java, etc) we had to engage the Jenkins or server admin. Every time we needed to update software (node, java, etc) we had to engage the Jenkins or server admin. Every time we wanted a new plugin we had to engage the Jenkins admin. Every time we needed to add server capacity we needed to re-setup the machine with all the build dependencies and software ...... you get the point. The worst part is that at large companies getting any of this done could take weeks and would slow down the development team. Perhaps even worse was when we couldn't upgrade software or plugins because the Jenkins servers were shared and other teams relied on older versions.
I believe Drone solves these problems because the developer has direct control over every aspect of the build process, from build environment to plugins, defined directly in the yaml file. If the developer defines a specific version of Node or Mysql in the yaml file the Docker image is automatically downloaded. If the developer defines a specific plugin the in yaml file the Docker image is automatically downloaded. We no longer have to worry about version conflicts between different teams because Docker images are tagged and isolated. We no longer have to worry about setting up new build servers because the required Docker images and plugins are sourced from your yaml and are automatically downloaded on-demand.
In addition, as @tboerger mentioned, I think Drone has a very unique plugin model that, in some ways, is more flexible and secure than Jenkins. This is because plugins are simply Docker images that are executed at runtime. Since a plugin is just a Docker image, you can easily write custom plugins in whatever language you want, and automatically distribute your plugin via the Docker registry. Your plugins can even auto-upgrade!
I think overall, if you use standard tools and processes and a Docker build environment, Drone will be easier to install and maintain thank Jenkins. It always surprised me how many companies had QA teams that, instead of testing software, worked as full time Jenkins admins. What a waste! This is why my goal is to create a developer-driven, developer-configured Continuous Integration system where you never need to login to a server, or ask a sysadmin for anything, ever.