The big difference is in the engine's ability to adapt to different situations and the amount of control offered.
Very simply, a carb works like a vacuum. When you open the throttle, the engine sucks in more air. As the air passes through the carb, the suction created (called a venturi effect) pulls more fuel with it, thus you get more gas as you step on the throttle. The amount of gas sucked at a given time is determined by the jetting of the carb & the number of jets. Typically a carb is tuned for a certain AFR at a certain altitude. If you change anything in the outside environment that would affect the volume of air your engine sucks (like driving to higher altitudes) it will affect your jetting. The car will still probably run but your AFR's will most likely be off. Carbs work (obviously, they've been around for a while) but the amount of control you have over them is limited to the jetting and the amount of air you let the engine suck in (usually the size of the carb, measured in CFM's). Carbs are also usually targeted to a very specific operating range, which is where they will be most efficient. There are some sophisticated carb setups where two jets are "on" for low RPM's and cruising, and when the throttle is floored two more jets are turned "on" (via mechanical linkage) to offer more fuel. It is very hard to get a specific AFR, and requires a lot of tinkering and fiddling with the jets to get it right - at a specific range.
Fuel injection is much more sophisticated. Essentially a computer measures the amount of air the engine just sucked in, and then tells injectors to squirt a certain amount of fuel. Because computers can measure and compute things very fast, they are able to measure the air intake in real-time and tell the injectors to squirt before the air makes it in the cylinder. Because the computer is measuring things, and is very good at measuring things, you have way more control over things like spark tables, the volume of the air, the temperature of the air, the amount of fuel used, etc. And to adjust things is messing with a computer, you do not need to take your carb apart to rejet it. Fuel injection is a lot more efficient (one of the main reasons engines get good gas mileage now) and a lot more adaptable to outside changes. Because FI is adaptable, the fueling is more efficient across a much wider range than that of a carb, and AFR's can be nailed pretty much dead on at many different operating ranges.
Some folks like carbs because they are simpler to deal with. Other folks like the adaptability and control that FI offers. Which one you want really depends on what you plan on doing with the car. 90% of the time I'd say you want to go FI. It seems more complicated initially but you can do a lot more with it. An example of the other 10% - NASCAR still uses carbs. So obviously they can go fast. But bear in mind that NASCAR cars basically run flat-out all the time. The carbs are set up for this. I'm sure these cars are much less efficient/fast (relatively) at less than flat-out. There are a number of reasons that NASCAR has kept carbs, one of them being the cars are already fast, swapping to FI will just make them faster. Think about that for a second.
In short: Carbs are 19th century technology. FI is 20th century and on.