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Fiberglass
A fairly generic term in which you take fiberglass mat or cloth and lay it up with resin to form nearly any shape. Found everywhere, boats, surfboards, airplanes, car bodies, rocket motors, space shuttle etc.
Pros- good strength to weight, easy to work with, can be made very flexible,low cost.
Cons- repeated flex can cause failure, gets brittle when uncoated/painted and exposed to sunlight. Glass fibers get in your skin when sanding and make you crazy.

Carbon Fiber
The same basic process as fiberglass only using carbon fiber cloth and high quality resins to get the highest tensile strength, highest strength to weight, and achieve a very rigid shape or structure.
Pros- Ultra high strength, ultra rigid, resists damage from sunlight when unpainted.
Cons- does not flex, if flexed it shatters like glass. very expensive.

Traditionally this was only found on ultra high tech applications (fighter jets, space shuttle, F-1 race cars, world class racing yachts) due to its high cost. It is now much more mainstream (even NASCAR bodies use carbon fiber now I think) and has become a fashion accessory of the Rice racers and others who have body and dash panels laid up with carbon fiber just for the look. Most of these panels are not pure carbon fiber but common plastic/metal panels with carbon fiber skin. No lighter in weight or stronger than fiberglass. Just pure bling. All foam...no beer.

So do you want carbon? Is ultra high strength/ultra rigid important for your application? If so then carbon fiber is the way to go if you can bear the extra cost.