Clay mitts and clay bars are two different methods to achieve similar results.
Clay bars need lubricant to pass over the surface of the paint without marring. Most require use of detailing spray as a lube, however Bilt hamber works using water. The key here is that as contaminants are picked up, you need to turn and fold the clay to capture them and reveal virgin clay material to continue the process. Failure to do so risks picking contaminants which can then damage the paintwork as the clay is moved across it.
Clay Mitts/clothes remove contaminants in a similar way, but need rinsing off to remove them from the working surface of the cloth. In this way they use the soapy water used to rinse the cloth as lubricant and don't need a detailing spray.
Just using detailing spray as a lubricant, and not rinsing the mitt correctly risks damaging the paintwork in a similar way to not folding the clay.
While in theory you could use a detailing spray as a lube for a clay mitt, you will still need to rinse it to remove contaminants, negating the need for the detailing spray in the first place.
I have used both clay cloth and clay bar, and have found that while the mitt is convenient for a quick decontamination after cleaning, the results of using the clay bar are superior. This could be down to my technique, as I adnit to taking more care using the clay bar, than wiping over with the cloth.
Regards, Neil