There is a difference in how they work, depending on if you are in the original renderer or the enhanced webGL renderer.


In the webGL enhanced graphics render the province labels button is in the settings menu. Note that they will only display when zoomed in, and will vanish when you zoom out to reduce clutter.







In the original renderer the province names are also tied to the resource display layer. So you can toggle the "display resources" button to show both resources and province names. You can also just hover your cursor over the market bar, and the resources and names will display temporarily.





So in the old renderer you can have a case where you have BOTH the resources turned on AND the province names turned on, so you may have to turn both of them off if you want the province labels to go away.