Take a look at these two pictures.
This is partly sunny.
This is partly cloudy.
Clearly, both pictures are the same. Beyond not being able to call cloud cover at night partly sunny, both phrases mean exactly the same thing: a sky with between 3/8 and 5/8 coverage of clouds. Although partly sunny may sound a bit more optimistic.
Mostly cloudy is 5/8 to 7/8 coverage, while mostly sunny indicates 1/8 to 3/8 coverage during the day.
But overcast and obscured skies are not interchangeable.
The sky is obscured when you can not see the sky because your vantage point is in the middle of clouds, smoke or fog.
An overcast sky is when the entire sky is above you is covered by clouds or smoke but you are not in them.