Sugar is made by repeatedly boiling and cooling cane syrup (or sugar beet syrup). After each cooling, the solution becomes supersaturated with respect to sucrose, causing sucrose to crystallize out of the "mother liquor" (the industrial term for the liquid solution from which crystals form). These crystals of "raw" sugar are heavily processed (washed in syrup, dissolved, treated with carbonate or phosphate minerals, recrystallized, and subject to ion exchange or charcoal color removal) to result in white sugar.
Nowadays, brown sugar is usually made by mixing refined white sugar with blackstrap molasses. In former times, the brown sugar was simply unrefined or unwashed crystal fraction from the 2nd or 3rd crystallization of cane juice. But precise control over the ratio of "contaminating" liquid to white crystalline sugar was difficult, which is why industry moved towards reconstituting brown sugar by mixing molasses and white sugar.
Blackstrap molasses is simply the spent "mother liquor" remaining after the third cycle of boiling and cooling to produce sugar crystals.
The smell of brown sugar thus comes primarily from molasses. The smell of molasses is very complex. Here is the abstract of a relevant paper:
The characteristic sweet-aroma components in refinery final molasses were isolated by using a combination of sensory evaluation, column chromatography and gas chromatography, and were identified by infrared spectrometry, mass spectrometry and color reactions. Identified were ethyl n-hexanoate, ethyl phenylacetate, phenylacetic acid, n-butyl benzoate, isopropyl benzoate, l(+)-pantolactone, benzoic acid, o-toluic acid, m-toluic acid, β-phenylpropionic acid, succinic anhydride, maltol, isomaltol, and 2-methyl-5-hydroxy-6-ethyl-γ-pyrone.
The aroma of the mixture of these compounds and vanillin and vanillic acid was similar to that of refinery molasses by sensory evaluation. The results indicate that these compounds are important in producing of sweet molasses aroma.