I'll admit I know very little chemistry, but I think the article would've been much better for people like me of it included any specific examples at all of benzene uses. It's filled with assurances that it's important and used all over the place but I didn't find that very enlightening.
It's kind of hard to explain how widespread a usage like "solvent" is. A huge amount of chemical reactions are most convenient to do in a liquid, a liquid that can dissolve the materials you're working with. That's a solvent. Benzene can dissolve a lot of things, including some that are really hard to dissolve otherwise. So it can be used for a huge variety of reactions, but as the sort background player that you might not pay attention to unless you can't have it.
It's a little like asking "what are the uses of water in chemistry", where you're tempted to answer, "um, everything?" Not quite, but not that far off either. (And with more cancer of course.)
Edit: disclaimer, I'm not a chemist, just an interested layman.
Benzene is used mainly as an intermediate to make other chemicals, above all ethylbenzene (and other alkylbenzenes), cumene, cyclohexane, and nitrobenzene. More than half of the entire benzene production is processed into ethylbenzene, a precursor to styrene, which is used to make polymers and plastics like polystyrene. Some 20% of the benzene production is used to manufacture cumene, which is needed to produce phenol and acetone for resins and adhesives. Cyclohexane consumes around 10% of the world's benzene production; it is primarily used in the manufacture of nylon fibers, which are processed into textiles and engineering plastics. Smaller amounts of benzene are used to make some types of rubbers, lubricants, dyes, detergents, drugs, explosives, and pesticides.
It's an important feedstock in the chemical industry but it is no longer used directly in household products. It used to be common in solvent/glue/grease remover formulations before the health hazard was widely appreciated.
Benzene was what made decaf coffee possible in 1905 (if can you consider this a productive use).
The beans were soaked in warm water then rinsed (several times?) with benzene, which was able to extract the majority of caffeine, and presumably not much else affecting the flavour.
It would have the benefit of evaporating with no residue given enough time, but due to the possibility of residue and the difficulty of working with it safely, decaffeination processes have since moved on.
