Track of the Wolf has Rice's 44" A weight .40 listed as 4.6 pounds, and the B weight .40 as 5.1 pounds. So it is likely the difference between a gun a little lighter than eight pounds and one between eight-and-a-quarter and eight-and-a-half pounds. If you go for a 38" it comes down to 4 pound barrel versus a 4.5 pound barrel, so say seven and a quarter pound, maybe a couple ounces heavier, versus seven-and-three-quarter or a couple ounces heavier.
None of these are exactly heavy guns, and personally I'd favor the heavier barrel for better steadiness off-hand, based on my experiences with a B-weight .50. Not to mention that 19th century stock designs were intended for barrels that were four or five pounds heavier....
(I note that Rice's listed weights and Track of the Wolf's listed weights do not jibe.)