Why we won't extract heat from air temperature? There is a lot of energy in air's temperature? Why don't you harvest it? There is always difference that creates flow which can be piped into something converting it to mechanical energy...
 A: Apart form what the other answers mention, also note that regardless of the energy conversion method, there is an upper limit for efficiency given by the Carnot efficiency:
$$\eta_{Carnot} =\frac{T_H-T_C}{T_H}$$
Efficiency is how much useful energy (such as mechanical work) you get out compared to how much thermal energy that must be absorbed. It is in general hopelessly inefficient to harvest energy from small temperature differences.
Comparing for example a $4\;^\circ \text C$ lake with ambient $34\;^\circ \text C$ air on a hot summer day will set the max upper efficiency limit at $\eta_{Carnot}=9. 7\;\%$. No machine is reaching it's max, so the realistic limit is far smaller. 
Efficiency is a big factor when determining which devices that are worthwhile to put effort, money and time into. 
A: We actually do. The difference in air temperature causes a difference in pressure. The difference in pressure causes winds which then power wind turbines.
A: We can only extract thermal energy from a temperature difference.
Since the air temperature is our surroundings, we can't  really extract the energy from it unless we had a lower temperature resovior.  See exergy for example.
Since we don't have something with large thermal capacity that is consistently cooler than the air, it's hard to extract the energy from the temperature of air.
On the other hand, temperatures drive all the wind energy in the atmosphere.  So in a way you can extract some of the thermal energy through wind turbines.
For low heating/cooling operations you can utilize a geothermal heat pump to take advantage of the temperature difference between underground and your surroundings.  Generally this is done between the ground source and your conditioned space though; and the outside air itself is not really taken into account (beyond heating cooling loads in the conditioned space).
The air itself is generally the target for the heating and cooling; so it's hard to use the air itself to provide energy for that process.
If you're talking energy generation, it's very useful to note that "low exergy" sources like geothermal produce very limited electrical energy, and are best utilized while kept in the thermal domain.
A: We do "harvest" heat from the air in a heat pump (eg air conditioning unit) but work has to be done for the heat pump to work.
That work done to make the heat pump work is unfortunately less than the work that can be abstracted from the different temperature heat reservoirs created by the heat pump.  
Nature does the job by having hot air (heated by the Sun) finding itself below cold air and creating convection currents.
So you now have moving air from which you can abstract (kinetic) energy.
