I believe the term “kubernetes operator” was introduced by the CoreOS people here
An Operator is an application-specific controller that extends the Kubernetes API to create, configure and manage instances of complex stateful applications on behalf of a Kubernetes user. It builds upon the basic Kubernetes resource and controller concepts, but also includes domain or application-specific knowledge to automate common tasks better managed by computers.
So basically, a kubernetes operator is the name of a pattern that consists of a kubernetes controller that adds new objects to the Kubernetes API, in order to configure and manage an application, such as Prometheus or etcd.
In one sentence: An operator is a domain specific controller.
Update
There is a new discussion on Github about this very same topic, linking to the same blog post. Relevant bits of the discussion are:
All Operators use the controller pattern, but not all controllers are Operators. It’s only an Operator if it’s got: controller pattern + API extension + single-app focus.
Operator is a customized controller implemented with CRD. It follows the same pattern as built-in controllers (i.e. watch, diff, action).
Update 2
I found a new blog post that tries to explain the difference as well.