The advantages for window cleaning are obvious, but for carpet cleaning I would say not.
I used to work for a water treatment company which produced "pure water" (RODI). The water production in the quantities needed for carpet cleaning would be very expensive for little beneficial return.
Theoretically, pure water could have the following benefits:
No scaling up of truckmount machines and tools
Removes chlorine and other impurities that could affect (wool) carpets
Ensures working at a neutral pH
Water will have predictable reactions with chemicals due to neutral pH and ionisation (not deionised water)
No residues
Marketing with an eco message
But all this is theoretical. The concentrations of chlorine and other contaminants are so low as to make no difference to using tap water. Those of us in hard water areas know the cost and inconveniece of limescale, especially when using a truckmount, but there are other cheaper and quicker ways of dealing with this problem.
Great question though and if anyone can make a good case for it I will be very interested.