In what way is a regular polygon less regular than a circle? I'm interested in finding a word (if it exists) for describing how regular polygons are less regular than circles. Even though they are regular, it seems apparent that there is some characteristic that separates them from circles. Basically, the points that make up the perimeter of a circle can be described by a single function instead of a piecewise function and all of these points are equal in all respects, whereas in $n$-gons, they become less "equal" with decreasing $n$. Specifically, I want to describe the difference described above between a hexagram and a circle _in one word_ (or maybe two). Right now, what I wrote reads "irregular" shape (referring to the hexagram), but that's just not right – a regular hexagram is not an irregular shape.

Similar to what Hagen said in a comment, the circle has infinitely many lines of reflection symmetry, while the regular polygon only has finitely many. Also, any rotation of the circle about the centre preserves it, whereas only finitely many rotations (modulo a full turn) of the regular polygon preserves it.

On the other hand, if you are talking about a parametrization of the shapes, then the circle has a smooth (differentiable) parametrization while the regular polygon does not. Another possible characterization is that there is no point equidistant from all points on the regular polygon, unlike for a circle.

Ultimately though, your question is too vague, but I can understand that perhaps you don't know the terminology to make it precise. Hope that one of the above is what you're looking for. If not, try to explain more clearly what you are looking for.