I know this thread is already two years old, but, while preparing for a path integration exam, I arrived at an intuitive picture that sheds some light on the origin of the extra term.
The picture represents an integral of a smooth function with respect to a concrete realization of Brownian motion. The sum of the areas of the green rectangles represents the difference between Ito (using the left point of each interval) and "anti-Ito" (using the right point of each interval) for sampling of the Brownian motion represented by the red line. Finer sampling leads to smaller rectangles, but they **overlap more and more** (because Brownian motion is not monotonic), so even if the area occupied by them tends to zero, the sum of their areas does not. This suggests (only suggests -- it is an upper bound on the difference, not a lower bound) that there is a "room" for Ito and "anti-Ito" to differ in their values. Stratonovich can be expected to lie somewhere in between.

Look at the following image:

https://lh6.googleusercontent.com/-bEPzm01WyGk/T-WplGQAc3I/AAAAAAAAACQ/mZr-5p0VUrw/s317/integral-wrt-brownian-motion.png