Believe it or not, there is a difference. It relates to the average amount of cloud cover expected. But would you believe partly sunny means more cloud coverage than partly cloudy? It can be confusing. Mostly sunny means more sun than partly cloudy does, while partly sunny means more clouds than partly cloudy does.
Growing up I wanted to become a Meteorologist. I was fascinated with the weather and dreamed of being the "weather girl" on television. That was until I became aware of the math requirements for the degree. Differential equations = different major! And I'm not a morning person. One does not begin their journalistic career on the evening news. You start on the morning edition around 5:00 a.m. Um, no thank you. But I still have a love for meteorology, enjoy learning about it, and find myself taking more and more pictures of the sky. I'll never forget my meteorology teacher at Mizzou. He was from Boston with a thick accent. He would say the sky in my photo is pahtly cloudy. He made meteorology fun.
So from least cloud cover to most, the scale is this: sunny, mostly sunny, partly cloudy, partly sunny, mostly cloudy, cloudy. Mostly sunny means more sun than clouds. Partly cloudy generally means an equal amount of sun and clouds like in my photo.
And now you know! Enjoy the weather.