Why do aircraft piston engines require priming?
Modern car engines are really very sophisticated compared to most airplane piston engines, which are 70 year old technology. 

Modern cars typically have an electric fuel pump in the tank which is often started by opening the drivers side door, by the time your butt is in the seat the fuel system is pressurized. When the ignition system is activated the ECU (engine computer) looks at engine coolant temperature and outside air temperature and directs the electronic fuel injectors to meter the exact amount of fuel which is correct for the conditions, directly into the inlet valves. A cold engine needs a much higher ratio of fuel to air when starting (2-3 times when hot), which reduces as it heats. When it starts running the ECU also takes into account the mass of the air flow and oxygen content of the exhaust to maintain the ideal fuel-air balance for the smooth and efficient running of the engine. 

This technology does exist for airplane engines, it's just expensive to fit so very few airplanes have it. Most light airplanes have very simple, completely manual systems. You shut down the engine by cutting off the fuel supply (closing the mixture), this ensures that the engine is completely dry of fuel so that if the prop is spun there's nothing in the cylinders which could go foom and spin the prop, potentially maiming someone. 

Piston engines create a lot of suction as they operate, and rely heavily on vacuum suction to operate. The low pressure in the carburetor atomizes fuel, which then gets pulled into the cylinder, explodes which in turn pulls more fuel and air in. However, because the mixture was cut off the engine and carb are all dry (any fuel left in the carb bowl will evaporate over time), which produces little suction. No suction, no atomization of fuel, no foom. The primer pumps a very rich atomized mist of fuel into the intake for each cylinder, giving the engine something to foom, creating suction to pull fuel through the carb and sustain the engine. This is why you feel resistance when using the primer, you are forcing it through a bunch of atomizing nozzles. This is why you need to start right after priming, if you wait too long you lose atomization.  

Note that many airplane piston engine carbs have accelerator pumps which jet raw fuel into the carb on quick throttle advances, this is to prevent a lean cut-off. Some people think it's the same as using the primer, but it isn't at all. Because there's no suction when you pump the throttle forward all you are doing is jetting liquid fuel into the carb, there's no vacuum to atomize it, so all you are doing is filling cylinders with raw fuel and they won't go bang. A single jet is good because having fuel in the carb bowl gives something to atomize once it catches, but too much will just flood the engine. 

A hot start is very different from a cold start, the engine doesn't need as rich a mixture and the carb bowl still has fuel in it, so when you crank it you don't need to prime in most cases.