Granulated white sugar is highly refined - that is why it is so white. Brown sugars have some residual amounts of molasses left in them - that is why they are brown. The brown solids are not digestible by bees, so they must be pooped out. In winter conditions when flying opportunities are limited, cleansing flights may not be possible, so the bees have to
hold it in.
In warmer weather when cleansing flights are not a problem, the solids in brown sugar
may not be that big of an issue if the syrup is consumed right away. But, it is always possible that the bees might store some of that brown sugar syrup for later use, so it might still get consumed later during a period when cleansing flight opportunities are limited.
The safest plan is to avoid feeding any sugar other than plain granulated
white sugar.
See this page on
nosema by Eric Mussen for some comments on the hazards of feeding alternative sugars:
http://entomology.ucdavis.edu/files/147621.pdf
Note that many sugars sold as "organic" have a brown tinge to them. The color is because those sugars are refined to a lesser degree than plain granulated white sugar. So feeding bees "organic" sugar that is
not bright white carries the same risks that are noted above.
.