Movement of air (the wind) is caused by difference of pressure. Air moves from the areas with high pressure to areas with low pressure.
Well, as a result the pressure decreases in the area the air moves from and increases in the area the air moves to.
As a result the pressure would become equal and the wind would stop. But how come Earth's atmosphere has not reached equilibrium during it's billion years history and from time to time the wind does blow?
Even if at some moment the atmosphere is in equilibrium, because the Sun heats the air not evenly, the temperature (and the pressure) of some volumes of air would change and air masses would start moving.
Now to your examples.
If you remove the wall in the first case (pressures are different), the gas will move very fast from left side to right side. There will be a kind of gust of wind - very strong, but short.
In the second example, nothing would happen at the moment you remove the wall. Left half of gas would gradually heat up, right side would cool down. By the time equilibrium is reached concentration of gas in both halves will become equal ($1.5 n_0$), so there would be some net movement of gas from left to right, but that's not a wind.