You're about to learn today.
Decisions are hard to make, especially when you're talking about what modification to put on your car. Should you use adjustable springs? How about progressive rate? Cross drilled or slotted brake rotors? How about what to decide when it comes to forced induction? This can be tricky, and there's no real right or wrong answer. The skinny is, they both force more airflow into the engine but in very different ways. Simply put, the supercharger runs off of the crankshaft and the turbo runs off of exhaust gasses.
Let's look at them more in depth though. With a supercharger, a belt is wrapped around a pulley and connected to the crankshaft. With this in mind the supercharger is spinning at the same speed as the crankshaft at all times. There's all this spinning, but what does it do?
If we're talking about a twin screw supercharger then it's two grooved rods that fit together spinning next to each other, acting like fans pulling air into the engine at a rapid rate. The good thing about a supercharger is that extra power is immediate. As soon as you hit the gas, the supercharger is throwing more air into the engine and your forced induction is showing results right away. The problem with this setup is that it's run off of the crankshaft. Power will be provided immediately, that's true, but at some point the fact that it's run off of the crankshaft means it'll eventually start taking power away from the engine. It has a peak effectiveness, which doesn't take very long to get to, and then it becomes just another accessory like an alternator.
Sure it's providing power, but because it's only going to be around 7 psi it's not the greatest amount of boost, and it will not last for too long. On the flip side, a turbocharger, which is run off of exhaust gas, takes much longer to build up power but it only does so if you want it to.
If enough exhaust goes through the compressor at a high enough pressure, the turbo will start to store it and run it back through the intake, thus recycling the unused fuel. Otherwise it would normally get dumped through the rest of the exhaust system. This spells efficiency. As for it being slower than a supercharger, there are ways around that but are significantly more expensive, such as a twin scroll turbocharger or a twin turbo setup. What manufacturers will do sometimes is tune the computer to keep one turbo spooled while the other builds pressure, each turbo handling one half of the cylinders. This is what BMW did with the E92 335i that used the N54 twin turbo charged engine. Now let's talk about application.
Since superchargers are so quick to delegate boost, the kind of racing that can really benefit from it is drag racing. You need immediate power and lots of it at low-end RPMs, and that's what a supercharger is good for. Turbocharging, however, is better for road courses. Lots of racetracks have one or two big straights, so there's plenty of room to build boost and spend it. Also because the boost isn't always there the problem of wheel spin isn't always there either. The down side is that turbocharging is way more complicated and expensive. So really, what are you going to use your forced induction for? Both have benefits and idiosyncrasies, but they both have their place with cars.
It's important then to do a ton of research on the engine that you are planning to put forced induction on. Either way, have a good time and prepare for potentially bad gas mileage, trees that appear as if from outta nowhere, and annihilating damn near anything on the road in a race.