The most obvious advantage of round shoelaces is durability. Flat shoelaces simply don’t last as long as round ones do because they get twisted up more easily and wear out faster under normal conditions. Not only do they work themselves loose more quickly, but they also fray at the ends more rapidly.
Round shoelaces tend to be stiffer and they retain their shape longer. The choice between round and flat shoelaces is primarily a matter of personal preference and the style of the shoes.