Answer

The cabinet of the K and KW loudspeakers is the most discernible difference. 

The K12 is made of durable and lightweight ABS plastic, while the KW122 is made of texture coated, premium-grade Baltic-birch plywood. The wooden enclosure reduces cabinet resonance frequencies which were EQ'd out from the K-series speakers, and returned to the KW to give a higher SPL (131 dB max SPL for the KW122, and 127 dB max SPL for the K12). 

Also, while the power amp of the K and KW series are the same 1000W module, the input module is different. The K12 offers a "MIC/LINE" switch for channel A depending on the type of input source, and the KW122 offers a "MIC/LINE" gain knob on channel A to select a 12, 24, or 36 dB increase to the input, allowing more versatility when working with a low-gain input sources. All other input/output connections and controls are the same.