The drawing of both the square knot and the granny knot contain two pieces of intertwined string (blue and red), each of which is roughly in the shape of a sideways 'U'. The blue string has the open end of the 'U' facing left, and the red string has the open end of the 'U' facing right. In tracing the overlap/underlap of the blue (left facing 'U') string, we start with the top part of the 'U' and find:

square knot: The blue string goes under, then over, then under the red string before rounding out the bottom of the 'U'. After the u-turn, the blue string continues to go under, then over, then under the red string to get back to the other open end of the 'U'.

granny knot: The blue string goes over, then under, then over the red string before rounding out the bottom of the 'U'. After the u-turn, the blue string continues to go under, then over, then under the red string to get back to the other open end of the 'U'.

The basic difference between the two knots is that they have different under / over / under windings for the first part of the 'U' but same under / over / under windings after the u-turn. I apologize if this description is difficult to follow, but the difference between the two knots is subtle and difficult for me to explain in words.

Updated July 1, 2003