"VDC" = Volts Direct Current, "VAC" = Volts Alternating Current. It seems like the Littermaid motor is designed to run on AC currrent, and apparently running it on DC, according to a previous poster (Dermanj on 12-24-03) eventually damages it.
It looks like the Littermaid company has simply boosted the amperage from 600 to 1000 milliamps in the new units, but continues to use AC rather than DC current. Based on what Dermanj said, and what Littermaid has done, it might be best to only use adapters that output AC current, and to limit the power to 12 Watts.
Voltage and current vary inversely, and power = volts * amps, so that 24VAC/500ma has the same power as 12VAC/1000ma, e.g.,
24 * 500 = 12000 milliwatts = 12 Watts
12 * 1000 = 12000 milliwatts = 12 Watts
I'm really not sure what the motors are actually rated for, but going above 12 watts and/or using DC current may not be a good idea as it may damage the motor.
If anyone is interested, a discussion of basic electrical concepts with examples can be found here:
http://www.elec-saver.com/e-primer.htm#Ohms'%20Law