What Steve said and I'll go a little deeper.
The kit gun .22's were intended as a "handy sized .22 for taking along while fishing, hiking, etc in your "kit". They started with the 22/32 kit gun in 1935'. Then onto the mod 1953 22/32 (22 cal on a 32 "I" frame). It was also built on an alloy frame as the mod 1955 . Later they evolved into the J framed steel mod 34 , (steel mod 35 target) and alloy mod 43. A mod 63 was also added which was esentially a mod 34 is stainless. These have all been discontinued and are followed nowonly by the 317 series in a alloy J frame . There are still lots of used 34's and 63's around though.
The K-22's used the larger K frame and also started prewar ( 31') and were known by K-22 . They were intended for target shooting, hunting and centerfire training. These eventually evolved into the model 17, 18 and lately the 617 (stainless version) which all have kept the K-22 designator in their titles. The 17 and 617's were K-22 Target masterpiece, the 18's combat masterpiece) As with Kit guns, there are still lots of good used K-22's to be had.
OK , bored yet? Bottom line; 22 kit guns were on the small frame. K-22's on the larger frame.
p.s. Info source is SCoS&W 2nd.
