I personally don't think they have much to do with one another. At the same time, however, there's a fine line between the two. A contradictory statement in some ways, but hear me out.
Common courtesy is an accepted extension of manners and etiquette often influenced by cultural environment. Our current environment of "F you and F you, too" has brought common courtesy to virtual extinction, but it's definition remains the same, nonetheless. Common courtesy is not practiced, in my opinion, out of any sense of political correctness; it's practiced as a result of personal morals. The two coincide with one another in some instances, but are totally at odds with one another in others.
For instance, if I hold a door for someone, especially an older person, it's considered common courtesy. However, if I hold a door or do a favor for someone specifically because they're a different race or ethnicity, that's political correctness. The former means I'm simply being courteous, the latter means I'm doing it to appease a different social or ethnic group. Here, we have an identical action that's being done for different reasons. I see it all the time.
Knowing this may be controversial, I won't go into too many different examples, but political correctness is often exercised out of racism and prejudice without the person even knowing it. If I make derogatory comments about someone of my own racial or ethnic group but am hesitant to make comments about someone from a different one SIMPLY because they're different than myself, no matter what that person did, that's the destructive behavior that's known as political correctness. In some cases, PC is called for and understood. In others, it's self-destructive, self-hating rhetoric that can only be surmised as prejudicial preference.
As an aside, I'm a very politically incorrect person. You know why I'm considered that? It's because I treat everyone equally. These days, treating everyone the same, with no preferencial bias, somehow seems to equate with racism and discrimination. I'm sick of it and won't change my ways, but that seems to be the manner in which I'm treated these days.
Long live common courtesy! To hell with political correctness.