Try and get the make and model number of the contactor so you can verify its specifications. If the contactor coil is truly 208 volts then you definitely have the wrong one. If you call for AC and go out to the condenser, you should hear a hum of the energized contactor. If it's a 205 coil I suspect you'll hear nothing.
Also you could take a pencil or insulated stick and physically close the contactor to verify there is the 240 volt power to the condenser fan and compressor.