Why would colder air disperse condensation on a car windshield? The outside temperature was somewhere between the hot and cold that blows on the inside through the car's vents. I know the condensate is on my side because I can reach forward and write on it with my finger. When I blow hot air on it, it stays there, but when I blow cold air on it, it disappears. I would have thought blowing hot air on it would give the water molecules enough energy to escape the surface and blowing cold air would cause water vapor molecules in the air to lose energy and get 'stuck' to the surface, but the reality is the opposite of what I expected. Can anyone explain?
 A: Condensation has to do with two factors, temperature and humidity. Specifically, in order for condensation to occur, you need a body of warm, humid air (humidity is necessary because otherwise there wouldn't be any water to condense) and a cold surface (the window), specifically colder than the dewpoint of the humid air, on which the condensation is to take place.
With a car window, this can happen on either side. In your case, you're getting condensation on the inside of the window because the interior air is warm and humid, but the outside air keeps the window cooler than the dewpoint of the inside air. So as long as you blow warm air on the window, you're just feeding it with more water to condense. In order to make the condensation evaporate by heating it, you'd have to be blowing enough air at a high enough temperature on the window to override the effect of the outside air and significantly heat up the window itself; your car's heating system probably isn't capable of doing that. (At least not while you're driving.) In contrast, as you pointed out in your comment, blowing cool air (cooler than the outside air) on the window means that the dewpoint of the inside air is necessarily less than the temperature of the window, so you don't get any condensation occurring. And as long as the air is less than 100% humid, any existing condensation will evaporate. This is especially likely if you're running the air conditioner, since it doubles as a dehumidifier.
The reverse situation occurs if it's hot and humid outside; you'll probably be running the air conditioner, so the cold air inside the car will cool down the window. Since the window is cooler than the outside air, you'll get condensation occurring on the outside. The solution in this case is to run the defroster to heat up the window and the condensed water molecules, giving them enough energy to evaporate.
A: I initially thought the existing answers were right, but after further thought and experimentation there are definitely some issues. I am open for discussion!

So as long as you blow warm air on the window, you're just feeding it with more water to condense

Heating air using a resistive heating element or thermal conduction from the engine block does not add moisture to it[1]. Where would this extra water to condense come from? 
Assuming David means it is absorbing water from the windshield but the amount of moisture in the vehicle is constant, if you run the "fresh air" mode that moist air is getting sent outside the vehicle. If you run in recirculate mode, yes, it will just redistribute the moisture to other windows and places in the car. 
There actually is a secret source of this moisture though that better explains your and my issues ... the A/C evaporator inside your dash that you used yesterday for that super-cool fast defrost mode[2]! That moisture is also picked up by the warm air, essentially pre-saturating it, and it condenses on the cooler windshield. Very few people notice this, and it easily foils mental models.
Lastly, the cabin heater on gas cars comes from the engine. On startup, it'll take a few minutes to warm up to have a larger temperature difference than the A/C. 

In contrast, as you pointed out in your comment, blowing cool air (cooler than the outside air) on the window means that the dewpoint of the inside air is necessarily less than the temperature of the window, so you don't get any condensation occurring.

Blowing similar or colder temperature air won't condense, yes, but it doesn't explain getting rid of the existing condensation on the windshield.
Cooling air doesn't magically get rid of its moisture. When cooling, any water vapor above 100% RH will either condense onto the object cooling it or turn into fog / a cloud. The now-colder air has less water than before, so when it's warmed up by the windshield it can "absorb" more water. 
How do we explain the quickness of evaporation relative to the warmer air? It has to be the A/C, which uses pressurized coolant to quickly move heat outside the vehicle. From looking around the internet, it appears that it will cool passing air by around 15-18 F [3], with no engine warm-up needed. Any moisture that is extra at that temperature condenses on the coils instead of getting sent as a fog out the vent (not exactly sure why yet, because I have seen this effect too), resulting in a faster defrost than the warmer setting. 
mring, you were right to mention vapor pressures at different temperatures, but I feel that is less of an effect and was confounded by the other variables at play.
[1] Combustion does create water though, which is why jet plane trails and visible car exhaust exist. Hot, humid air from gas exploding hitting air below its dew point.
[2]: Which is why I would recommend only using it when you have to. My morning inside-of-the-windshield condensation issues went away when I stopped using the A/C to dehumidify the car quickly and instead relied on the slower effect of drier outside air hitting the dash. If it's raining then ... I haven't thought enough about the perfect answer yet :-)
[3]: https://diy.stackexchange.com/questions/44203/how-should-i-evaluate-my-central-a-c-to-see-if-its-working-properly. Side note, humid air will cool much less than drier air, as the heat of condensation of water is 2,260 kJ/kg but the specific heat (the amount it takes to cool it one degree without phase change) of air and the liquid water in the air is 1 and 4 kJ/kg. 3 orders of magnitude! In this case it doesn't matter, as any energy-expensive condensation occurring on the A/C is going to be balanced by the necessity to evaporate water from the windshield, but it's good to know that your A/C won't feel very cold in a very humid environment.
A: Please correct my answer if its wrong:
I think that condensation of water inside the car has to do with taking into consideration the temperature and humidity outside and inside your car.
You only get condensation whenever its colder outside, I think.
Outside is colder than Inside:
So, I think that your car acts as an insulator of heat. Your car's air temperature inside is hotter than the outside, and therefore water is evaporating and expanding in all directions. However, there's no where for the water to go, so it builds up on your windows.
If you blow hot air, you'd end up heating the inside car's air temperature even more and end up with more condensation. If you blow cold air, you make the inside car's air temperature to be closer to the outside and therefore the water molecules have less energy and won't expand out in all directions and end up condensing on your windows.
I think that if you open your windows and let the inside temperature and outside temperature of air to come to equilibrium, you'll end up with no condensation on your windows from the inside.
Please test this and tell me if I'm wrong. Thank you.
