To me, stateful UI means that the front end is using variables to manage its behaviour.
For example, I'm working on an application that is transitioning some of our stateful navigation to be more stateless to hopefully address some end-user confusion. We have an overlapping hierarchy where many objects have multiple parents, and the same object would display different breadcrumbs depending on which path someone took to the object. This made people confused about whether they were looking at the same object.