Why is there no temperature difference in the Joule expansion experiment? The whole system is adiabatic, and no heat exchange can take place. If the volume of the gas now doubles, it should actually cool down.
That's why I don't understand $dT=0$
 A: 
if i have a piston and i pull on it really hard and this expands the isolated gas, i have no temperature difference?

In the case of the moving piston, the molecules of the gas are striking a moving wall.  The pressure of the gas creates a force and the motion of the wall means this happens over a distance.  So work is being done.  This work comes at the expense of energy in the gas and it cools down.
In the Joule expansion, there is no moving wall for the gas to strike.  There is nowhere for the energy to leave, so the temperature remains constant.
A: I like to think about these kinds of thermodynamic problems using kinetic theory and Newtonian mechanics, and not really worry about the ideal gas equation. If we look at problems this way, then temperature changes are really easy to understand. For example, let's consider the classic example where you adiabatically compress a gas in a cylinder. Kinetic theory tells us that we can envision a bunch of tiny point masses bouncing around with random velocities. As the shaft of the piston moves downwards, it collides with some of the molecules, and because of the equations for momentum conservation, this means that the gas molecules will ricochet off of the shaft with a higher velocity than when they entered, so they'll move faster and have a higher temperature. The same is true for when you pull a piston shaft upwards, allowing the gas to expand. The gas strikes the shaft as its moving upwards, and thus they ricochet off with a smaller velocity than when they entered. In Joule expansion, there's no momentum changes nor forces exerted on any individual molecule, and therefore there speeds can't increase or decrease, so it's the same temperature.
A: 
The whole system is adiabatic, no heat exchange can take place

The system also has no work done on it. The idea is that you just remove a partition and the gas freely expands. This is what defines Joule expansion. No heat and no work. Therefore, no internal energy change and no temperature change (for an ideal gas).

if the volume of the gas now doubles it should actually cool down.

No, this is wrong. An ideal gas will not "actually cool down" during Joule expansion. The volume doubles, but the pressure halves.
So, to answer your question:

Why is there no temperature difference in the Joule expansion experiment?

It is because no work is done on the gas and no heat is exchanged with the gas, therefore the energy is constant. The energy of an ideal gas is $\frac{3}{2}NkT$. The particle number $N$ is constant, so the temperature of an ideal gas is constant in Joule expansion.
A: We now can detect is a slight change in temperature of the air used in the Joule experiment that Joule was unable to detect.  This is because a gas is not an ideal gas.  Assuming the gas is ideal, there is no change in temperature.
Consider the air on both the initial and final chambers as the system, a closed thermodynamic system (constant mass). Joule observed no change in temperature of the surrounding water bath and concluded no heat transfer had taken place to the air.  Since the work done was also zero, he concluded from the first law of thermodynamics that the internal energy of the air was constant.  Since the pressure changed, he concluded that the internal energy of a gas is a function of temperature alone; this is true for an ideal gas.
A couple of additional observations based on other responses and comments
The gas is expanding into an (essentially) vacuum with zero pressure and therefore does no work; this is free expansion and is not the same as expansion against a piston with a pressure on the other side.
Considering the initially empty chamber as a fixed volume system (open thermodynamic system), there is no work done on the fixed volume system as the gas expands to fill it.
A good textbook on thermodynamics, such as one by Sonntag and Van Wylen, evaluates the Joule experiment from three points of view: (1) air on both the initial and final chambers as the system, (2) gas expanding into a vacuum with zero pressure, and (3) considering the initially empty chamber as a fixed volume system.
A: The local cooling throughout the ideal gas taking place during this irreversible expansion is exactly offset by the local viscous heating of the gas (viscous stresses combined with high velocity gradients).  For an ideal gas, the net effect is no temperature change.  See a course in fluid mechanics.
