With a carburetor, fuel 'flow' is coupled to air flow: basically, you cannot stop the fuel supply completely. (Ok - you can. But only when adding a stop valve.) And at low air flow rates you cannot get high fuel rates.
Fuel injection is decoupled from air flow: if you want to stop the fuel supply, just deactivate the injector. And if you like to have a very 'fat' mix: inject more.
Overall, fuel injection allows for a much better control of the fuel supply. Including cylinder-specific control schemes.
Whether fuel injection will replace the carburetor? It already has. At least in passenger cars in Europe and the US. Given exhaust gas legislation, no car without fuel injection is capable to pass the acceptance tests.
This is likely to be different in regions, where 'simple and cheap' vehicles have still their right to exist. And the carburetor may 'survive' in simple 2- and 4-stroke applications (2-wheelers, chain saws, lawn mowers etc.) where simplicity prevails fuel consumption and exhaust gas concerns.
Fuel injection (FI) systems provide better possibilities to consider environmental conditions, e.g. ambient temperature, air pressure (as a function of the altitude). This benefits to customer handling, e.g. in case of cold start behavior and engine drivability. In addition, the more accurate control of mixture preparation across the entire engine operation map supports better exhaust emissions.
One drawback of FI systems is the necessity of an electric power supply, e.g. by a battery. In some applications, this is not given, e.g. in small handheld tools or motorcycle engines. In addition, emergency start possibilities are limited, if the battery is not working. This point is topic of research for vehicles that are operated in outback conditions, e.g. in desserts or in far north countries. As special solutions, some FI system supplier have developed technologies for small and midsize engines, that can be operated without a battery. Starting is enabled by hand- or kickstarter; and the (small) energy demand of the FI system during starting is provided by a specific electrical system.
I would like to tell about the major disadvantage of the carburetor system by adding to the statement provided by U. Dreher and Mario,
Carburetors are well suited for low-speed single cylinder small engines, where there is no requirement of instant high power. These engines are used in mainly in scooters and some small motorbikes. These vehicles do not run at very high speed and do not meet the requirement of instant power.
But if it is required to use a carburetor in a multicylinder engine, say >3 cylinders, then this system fails to provide an equal amount of fuel in all the cylinders. Why does it happen?
Assume a case of a four cylinder engine. In the carburetor, the fuel molecules are atomized in the form of a fine spray and are carried by the suction air going into the combustion chamber. The cylinder which is located just adjacent to the carburetor gets a homogenous and stoichiometric supply of fuel+air. When the charge (fuel+air) has to travel some distance to go to the farther located cylinders, i.e 2nd and 3rd cylinders, the fuel molecules combine with each other and form a bigger molecule, which has higher momentum in comparison to the smaller ones. Due to greater momentum of the combined molecules, and high-speed flow of charge, these molecules are unable to retard to go into the cylinder (2nd and 3rd) and due to the inertia of motion, they reach at the end of the distribution rail. Now the fuel which was to go in the 2nd and the 3rd cylinder gets collected at the suction point of the 4th cylinder, as a result the 2nd and 3rd cylinders get a lean mixture whereas the 4th cylinder gets a rich mixture (fuel of 4th + some fuel of 2nd + some fuel of 3rd). So the multicylinder engines operating on carburetor systems, have unequal charge distribution and hence a significant fluctuation in the power output is also recorded.
This the main reason why carburetor systems are not preferred for multicylinder engines. Like Mario told, there many other issues related to the altitude also.
But in the case of FI (Fuel injection) systems, the fuel is injected just before the inlet manifold and is carried into the cylinder by the incoming air. This ensures a calculated and equal supply of fuel to all the cylinders thus resulting in better combustion, better efficiency and constant power output. No doubt it consumes some power from the battery but, to a great extent compensated by the higher efficiency of the system.