What advantage does a 4-in. octagon box have over a 3¼-in. octagon box?

Is that the whole question?  What kind of advantage are we talking about here?

One possibility would be that the 4-inch box has a greater volume (capacity) than does the 3 1/4-in box.  If that's the question here, then your job  is to calculate the difference in volume.  Find the vol. of each box and subtract the smaller from the larger to calculate the "size advantage" or "volume advantage."