Why are engines with carburetors hard to start in cold weather?
And why is there a different procedure for hot start scenarios, after you've just shut down the engine and want to power up again?
They are all harder to start in the cold, fuel injected engines included, and need an over-rich mixture for starting. The rich mixture is required because a lean one is harder for a spark to light off if the fuel/air charge is cold.
Now, it's still cold outside after the engine warms up, so the air charge is still cold, initially, but on a warm engine it gets heated as it extracts heat from the intake passage, valve and cylinder head on its way into the combustion chamber (and on Lycomings, the carburetor, which is bolted directly to the oil sump).
Carbureted aircraft engines use primer systems instead of chokes, as mentioned in AEhere's answer, because it's undesirable to have that sort of obstruction device in the intake passage. The primer injects raw fuel at the intake port using a syringe type hand pump, most of which settles on the intake passage walls and runs down the tubes a bit, and sits there slowly evaporating, and when the engine cranks, the vapours add to the fuel normally drawn at the carburetor. It does this for a period of time after the engine starts depending on how much prime you used.
Fuel injected aircraft engines are primed the same way, but since there are injectors already there at the intake port they don't require a separate primer system. You just turn on the fuel pump with the mixture rich for x seconds and it injects raw fuel just like a primer system.
Carbureted aircraft engines also use a slightly richer than necessary setting in the idle circuit to help them run better when cold. That's why you get a little rpm rise when pulling idle cutoff.
You can have too little primer fuel when starting, or too much, and it varies widely depending on the engine, its temperature, and the air temperature, so you have to learn how much is enough on each engine type. A typical protocol is:
Different engine models have different starting quirks, and different primers have varying fuel volumes per stroke. It's something you learn on that engine type, along with whatever other unique characteristics of a particular aircraft you have to be aware of.
Carburetors don't work as well in cold conditions because they themselves tend to cool the air as it flows through the Venturi tube, compounding the issue of cold intake air and cold fuel.
When the engine is cold, fuel vaporizes less readily and tends to condense on the walls of the intake manifold, starving the cylinders of fuel and making the engine difficult to start; thus, a richer mixture (more fuel to air) is required to start and run the engine until it warms up. A richer mixture is also easier to ignite.
To provide the extra fuel, a choke is typically used; this is a device that restricts the flow of air at the entrance to the carburetor, before the venturi. With this restriction in place, extra vacuum is developed in the carburetor barrel, which pulls extra fuel through the main metering system to supplement the fuel being pulled from the idle and off-idle circuits. This provides the rich mixture required to sustain operation at low engine temperatures.
To put it in simple terms, carburetors create a combustible fuel air mixture by letting fuel evaporate into a stream of air in more or less controlled conditions. To do this, they reduce the pressure of the air by forcing it through a Venturi tube; this facilitates fuel evaporation.
An undesirable side effect of the process to reduce the air stream pressure is a drop in temperature. A lower temperature makes it harder to vaporize the fuel, and if the intake air and fuel are already colder than usual, the resulting mixture might be too lean to burn.
This is countered via special cold start devices such as chokes, manual mixture control and heaters. Of course if the engine is already hot, having been shut down recently, it might be able to restart without such contrivances. Keep in mind, however, that the carburetors will usually cool much faster than the engine block, being smaller often exposed to more airflow, and all it takes is for them to cool down to impede a restart.
Hot starts are sometimes difficult on fuel injected engines because the engine heat will boil the fuel in the fuel lines, creating a vapor lock. Fuel pumps are designed to pump liquid fuel, not fuel vapor. The same is true for carburetor engines, although usually not as severe.
Look at your POH hot start procedure. Usually this involves running the fuel pump to move liquid fuel into the lines, flushing out the fuel vapor, and in most cases flooding the engine as well. Then the flooded start procedure must be used.