Both guns are .38 special only models on the S&W "K" frame.
One gun was intended for the target shooting market while the other was for the law enforcement market.
The Model 14 K-38 Target Masterpiece was, as it's name implies, S&W's premium "target" model revolver. It was offered with a 6" barrel (and later an 8 3/8" barrel) with partridge front sights and the S&W Micrometer adjustable rear sight. Triggers were serrated types with either the .265" wide or .500" wide trigger with overtravel adjustment. Hammers likewise were either width. Most were blued and nickle specimens are considered rare. Produced 1947-1982.
The Model 15 Combat Masterpiece was aimed at the law enforcement market as the name implies. While similar in appearance, it was originally offered only in a 2" or 4" barrel with a Baughman Quick Draw front ramp sight and S&W Micrometer adjustable rear sight. Triggers can be a serrated semi-target trigger or smooth combat trigger. In 1986 a 6" and 8 3/8" barrel was added to the available options. Available in blue or nickle finishes. Produced 1949-1999.
One could look at these guns as the same model with different barrel lengths and minor cosmetic changes. However the action on the Model 14 was typically much smoother from the factory than on the Model 15. Many rural police agencies and highway patrols equipped their officers with the 6" Model 14 instead of the shorter Model 15.