Can you elaborate a bit on what you mean when you say "integrate OpenShift with docker"? OpenShift is already tightly integrated with docker in its default configuration (Docker images serve as the image format, Docker engine runs the containers on the OpenShift hosts, even the integrated registry that is deployed by default is a Docker registry).
As for why you should pick OpenShift vs Kubernetes, I would point out that OpenShift incorporates additional features beyond just the container orchestration provided by the kubernetes components (Developer tools, Application Services, Cluster Services like integrated cluster metrics & log aggregation, etc).
Additionally, you are getting enterprise support for an open-source project (incl. security auditing, security patching, certifications and documentation support for software accreditation, support teams to respond to issues, enterprise training to support the skillsets needed to leverage the software, etc). The enterprise support is less important if you are an individual looking to do something relatively simple with containers, but in a large, complex business environment with mission-critical applications, the support is critical for innovation, application stability, and success. Few organizations have a mature enough IT/engineering org to justify maintaining their own implementation of the open-source kubernetes software in a secure, stable, and efficient manner.
For further reference, there was actually a recent discussion on this forum about the differences between OCP and Kubernetes (with some good links):