It is absolutely wrong to compare Docker and Jenkins, its like comparing cat with a dog. The ultimate purpose of Docker and Jenkins is entirely different, so you cannot do compare it. Below is the list of things you need to know about Docker and Jenkins.
Docker is a container engine that can create and manage containers, whereas Jenkins is a CI engine that can run build/test on your app.
Docker is used to build and run multiple portable environments of your software stack. Jenkins is an automated software testing tool for your app. Also, don't forget that Jenkins started as a build tool for Java apps.
Jenkins provides an easy interface to create and manage multiple CI/CD pipelines using its plugins. Sometimes people build Docker images and run Docker containers using Jenkins CI/CD pipeline for software testing purposes.
You can easily trigger in-server/in-container deployments based on Chef, Ansible, and Salt using Jenkins CI/CD pipeline.
We extensively use Docker containers for running SloopEngine environments. We have also learned how to implement and use Docker with Jenkins in a proper way. SloopEngine is a cloud-based SaaS product that offers key features such as server-backed online IDE, project management, centralized SSH key management, server access control, SSH logging & auditing, live SSH auditing, activity tracking, live chat, and more for tech teams.
Hope it helps. Thank you.