MIKE MOSS SAYS: Susan, Both of those terms represent a mix of clouds and open sky, ranging from about three to five eighths (or "oktas") of the sky covered by opaque clouds, and the National Weather Service officially defines the terms interchangeably. However, some forecasters lean toward using "partly sunny" when a bit more cloud cover than open sky is expected, and vice versa for "partly cloudy."
In addition, one would use partly cloudy rather than partly sunny for nighttime skies that fall within that range of cloudiness. Likewise with "mostly clear" versus "mostly sunny," both of which refer to around 1-2 oktas of cloud cover.