Brown sugar and Turbinado are 2 different animals. So this makes a flavor difference too.
Brown sugar, is refined white sugar with molassas added 3-4% for light and 6-7% for dark. All sugar in its rawest state has molassas on it but it is removed in the refining process.
Turbinado, (or Raw sugar or natural Brown sugar), is brown sugar where some of the natural molassas is retained instad of spun off during refining. the percentage of molasas left in turbinado is substantially less than in the lightest Brown sugars. The molassas makes the brown sugars more robust/complex in flavor compared to the sweeter turbinado.
Its the amount of molassas in the 2 sugars that come into play in our craft.. Turbinado will hold up to the longer heat exposure in butts and brisket where a brown sugar will begin to carmelize much sooner in the cook. Brown sugars on the other hand when used on shorter cooks like ribs and chicken will carmelize in the 4-5 hours where turbinado may not unless you pump the temps up.