The advantage to thick top bars is preventing burr between the boxes. The advantage to thin top bars is much more comb in the box.
“…that very practical Canadian bee-keeper, J.B. Hall, showed me his thick top-bars, and told me that they prevented the building up of so much burr-comb between the top-bars and the sections…and I am very glad that at the present day it can be dispensed with by having top-bars 1-1/8 inch wide and 7/8 inch thick, with a space of 1/4 inch between top-bar and section. Not that there is an entire absence of burr-combs, but near enough to it so that one can get along much more comfortably than with the slat honey-board. At any rate there is no longer the killing of bees that there was every day the dauby honey-board was replaced.”--C.C. Miller, Fifty Years Among the Bees.
“Q. Do you believe that a half-inch thick brood-frame top-bar will tend to prevent the bees building burr-comb on such frames, as well as the three-quarter inch top-bar? Which kind do you use?
A. I do not believe that the one-half inch will prevent burr-combs quite as well as the three-quarter. Mine are seven-eighths.”--C.C. Miller, A Thousand Answers to Beekeeping Questions