This is a picture of whole cane sugar, adapted from a photo provided to Wikipedia by Fritz Lasinger. First thing to notice is that it looks NOTHING like that "Sugar in the Raw" stuff, which is highly processed stuff, not much different than "brown sugar" which is white sugar with some molasses sprayed on.
IMO, there are three considerations with sugar in addition to the processing issue: glucose content, fructose content and practical use.
Beginning with the processing issue, most white sugar is made from sugar beets, most of which are GMO. White cane sugar is not GMO, whether organic or not, as there isn't any GMO sugar cane yet. So you can avoid GMOs just by buying cane sugar.
In processing the white sugar, a side product is molasses, which is basically all the minerals from the raw sugar. Most of your standard brown sugars are highly processed white sugar with some molasses sprayed on for color and flavor.
The primary processing issue is that the minerals are removed. However, I do not feel this is very significant, as anyone who is getting significant mineral content from their sugar intake has way bigger problems to worry about than their mineral intake. I do feel there's a much more important reason to use natural sugars, which I'll get to when I discuss practical use below.