What's the difference between OpenShift and Kubernetes and when should you use each? I understand that OpenShift is running Kubernetes under the hood but am looking to determine when running OpenShift would be better than Kubernetes and when OpenShift may be overkill.
In addition to the additional API entities, as mentioned by @SteveS, Openshift also has advanced security concepts.
This can be very helpful when running in an Enterprise context with specific requirements regarding security. As much as this can be a strength for real-world applications in production, it can be a source of much frustration in the beginning. One notable example is the fact that, by default, containers run as root in Kubernetes, but run under an arbitrary user with a high ID (e.g. 1000090000) in Openshift. This means that many containers from DockerHub do not work as expected. For some popular applications, The Red Hat Container Catalog supplies images with this feature/limitation in mind. However, this catalog contains only a subset of popular containers.
To get an idea of the system, I strongly suggest starting out with Kubernetes. Minikube is an excellent way to quickly setup a local, one-node Kubernetes cluster to play with. When you are familiar with the basic concepts, you will better understand the implications of the Openshift features and design decisions.
OpenShift includes a distribution of Kubernetes, so if you don't need any of those added features of OpenShift you can choice to ignore them such as: Web Console, Builds, advanced deployment models and much, much more.
Here's a summary of items available on the OpenShift website.