I understand that the venturi speeds up the flow of fuel and air which atomizes the fuel, but I don't see where vaporization happens. Atomization means the breakup of liquid into fine particles while vaporization means a transition from the liquid phase to vapor. Are both of these processes happening simultaneously? Does vaporization give the fuel molecules even more surface area than atomization?
are both processes (atomization and vaporization) happening simultaneously
Yes, the fuel enters the intake airstream of the carburetor as a liquid. The air flow breaks the fuel into smaller droplets. Evaporation will be ongoing as the fuel travels through the intake manifold toward the cylinder(s). Ideally, fuel and air are in a gaseous state before spark ignition. Higher temperatures in the cylinder ensure this.
This brings us to the question "why is a cold engine harder to start?".
Fuel needs to be within a certain % vapor range to burn, known as the lower and upper flammability limit. If it is very cold, not enough fuel evaporates to start the engine. This is solved by closing or "choking" the air intake enough (richening the mixture enough) to reach the lower flammability limit. Another solution is to inject a more volatile substance, such as ether, to get the engine started until it "warms up".
Mathematically, evaporation rate of fuel such as 100LL is time and temperature dependent:
ln(% evap) = (0.5 + 0.045Temp) × ln(time)
So, we can see this question has many variables, but most importantly, at lower throttle settings, the cooling effect of fuel evaporation can cause icing of the carburetor. Carb Heat ON helps prevent this while descending at a lower power setting.
Atomized and vaporized mean basically the same thing, though as you noted, "atomized" is usually defined as very small droplets, and "vapor" as something that includes both very small droplets and gas.
Hence it's probably better to ask if the venturi creates an atomized spray (i.e. very small droplets), or the liquid undergoes a phase change to a gas.
And the short answer is: yes. That means that both are taking place.
When the engine is operating normally, the venturi draws in fuel and it's atomized at the nozzle (i.e. turned into very small droplets). However, the vapor pressure is in most cases such that most of the components of the fuel undergo a phase change to a gas.
Having liquid fuel turn into its gaseous state in a venturi is not advantageous for the combustion process. That has something to do with keeping the combustion controllable among other things.
This is why inlet air is not preheated more than needed to keep it from freezing up. The inlet air however is still a gas and some of its surplus energy content is unavoidably shared by the fuel in the venturi, resulting in some of the fuel actually evaporating, i.e. transitioning into a gaseous state.
In LPG engines, the liquid fuel is vaporized before it gets mixed with the inlet air. In natural gas engines the fuel comes out of the tank as a gas, so it can't vaporize any more. In general, burning gaseous substance instead of liquid, makes for a hotter but less powerful engine.