They are similar, but not the same.

Some Similarities:

They both have squares in the corners and triangles on the sides.

They both have parallelograms with 45-135-45-135 angles for the centers.

Some Differences:

The proportions are different.

The shapes of the parallelograms are different.

The LeMoyne Star - all the interior (finished) seam lengths are the same.

The 8-pointed Star - The interior (finished) seam lengths are different.

The LeMoyne Star - all four sides of the diamond are the same length.

The 8-pointed Star - opposite sides are the same length, but not all four sides are the same length.

The LeMoyne Star - Cannot construct the center from HSTs