What's watts, watts is the work being done, watts for the 110v appliance can be the same work being done as the work being done by a 220V appliance, difference? A X B = C, when C does not change but A increases, B decreases.
I the old days the 6 volt electrical system wires were bigger and connections ran hotter, in 1955 the electrical system went to 12 volt, the watts required to start the engine did not change, Volts doubled from 6 volts to 12 volts, amps dropped 1/2 and the work (watts) stayed the same. Increasing the voltage and reducing amps (current flow-EMF) allowed auto makers to reduce the size of wire, they saved money by increasing voltage.
110V motors are bigger than 220V motors when the rating in watts is the same, 440V/3 Phase motors are smaller, cost less to run and do not require large wires.
To answer your question the work being done must be known for both appliances before a comparison can be made. If necessary I would add a plug next to the dryer plug for an additional 220 plug, I would avoid a pig tail adapter, I do not like disconnecting the plug on dryer without flipping the braker, to me it seems like there is always a chance of getting the fingers between the dryer connection and wall plug.
F. Guffey