I don't think I've ever heard partly sunny. Here's my take on them though.
Mostly sunny tends to mean how much of the day is going to be sunny, as in it'll be sunny most of the day, but then clouds will come in for a few hours, either as they're rolling in later in the day or rolling out earlier in the day.
Partly cloudy can mean you'll see clouds all day but it won't be full coverage. Or possibly light clouds that full cover but let a lot of sunlight through. Partly sunny would more or less be the same thing as this, to me at least. Its not a gloomy day, and/or the clouds are broken up such that there's a quite a bit of clouds but the sun will shine through often
Mostly cloudy would be like mostly sunny in that it'll you'll have full cloud coverage for most of the day, but the sun will come out at some point (usually either in the morning or evening as the clouds either roll in or roll out of the area).
Then again, I don't generally pay much attention beyond it saying its going to be sunny/windy/stormy and what temps. The rest is fairly irrelevant as far as what I need to know about the weather.