jmc

The sum of the angle measures in a polygon with $n$ sides is $180(n-2)$ degrees. So, the sum of the octagon's angles is $180(8-2)=1080$ degrees. The polygon is regular, so all the angles have the same measure, which means each is $\frac{1080^{\circ }}{8}=135^{\circ }$. Similarly, the sum of the angles of a hexagon is $180(6-2)=720$ degrees, which means each angle in a regular hexagon has measure $\frac{720^{\circ }}{6}=120^{\circ }$.

Therefore, the desired difference is $135^{\circ }-120^{\circ }=\boxed{15^{\circ }}$.