This is a trick question. There really isn't much of a difference. They all mean the sky is part clear, and part cloudy. It just gives us a larger vocabulary to work with.
Although most of the time, we'll tend to use it as a bit of psychological factor. If the forecast is improving (as in, rain event is just ending and we'll clear up a bit), we'll usually go partly sunny.
If the forecast is deteriorating (as in, it's sunny now, but rain moving in tomorrow), we might say mostly cloudy as the condition during the increasing clouds.
We tend to save 'partly cloudy' for nighttime, since we can't use partly sunny.