The difference is "host opens a goat door because he knows which one has the car and chooses to not open it" versus "host opens a goat door by random chance".

For the purpose of the problem there's no difference between the two cases. The only thing that matters is that the host (whether by knowing or by chance) opens a door with a goat behind it.

There is a big difference, "Ignorant Monty"/"Monty Fall" makes switching 50/50 instead of 67/33. The main change is that the door could have had the car, but didn't by chance, and we get to ignore the (one third of the) times he would have revealed a car by mistake. The original Monty never shows you the car because he always picks right, so those possibilities could never happen.

That the host opens a door without prize.