This is just a minor issue, but I got someone asking me about this, so:
If I search for something via the widget and press return, I get to the search result page. When I now click on the search widget again, it shows the search term and a spinner but I’d guess I does not actually load anything.
Also, when I change the search term in the full-page-search text field, the search term in the widget stays the same. Not sure if this is actually a problem. It looked a bit confusing.