What does adding butter to toffee do

Butter largely acts to prevent the crystallization of sugar in toffee. This leaves the toffee smooth and breakable.

Fats in candy serve a similar purpose. Fatty ingredients such as butter help interfere with crystallization—again, by getting in the way of the sucrose molecules that are trying to lock together into crystals. Toffee owes its smooth texture and easy breakability to an absence of sugar crystals, thanks to a large amount of butter in the mix. There's more about the role of ingredients in candies here.

You should still be able to make stickjaw (or, if not authentic stickjaw, at least a more sticky) toffee. The consistency of toffee has more to do with the temperature you bring the toffee to while cooking. Here's a chart detailing the various stages of candy temperatures. For a sticky toffee you'd most likely want to bring your toffee to the soft-crack stage.

Here is a recipe for authentic, old-fashioned, stickjaw toffee but without butter.