Translation:That popular singer does not have a car, but an airplane.
Not really, no. "Nincs" replaces "nem van" when those two would stand together. But, as you can see, there is a car between the two. And it has to be there, because the car is what is being denied: Not a car but an airplane instead. "Car" must stay right next to "nem".
So, we can't really stick a car inside the word "nincs". We have no choice but to use the two-word equivalent "nem ... van".
No, it is not.
"Nem zöld ruhája van" - what she has is not a green dress.
"Nincs zöld ruhája" - she does not have a green dress.
Different things are negated in the two.
To make it clearer in another example:
This is not icecream: "Ez nem fagylalt."
There is no icecream: "Nincs fagylalt."
Oh, that makes sense. In the first example, "green dress" is being negated, and in the second, "have" is negated. Yes? So if you're negating "have," you have "nem van" which equals "nincs." I guess the first sentence could be translated as "it's not a green dress that she has."
And there is an important difference with the "but instead" situation, as well:
"Ez nem fagylalt, hanem palacsinta." - This is not icecream but palacsinta.
"Nincs fagylalt, de van lángos." - There is no icecream, but there is lángos.
Jó étvágyat! :)
Update: I have to mention that there is an, I think, archaic usage of "hanem", which is now mostly replaced by "de" or "viszont".
"Nincs fagylalt, hanem van palacsinta." I think this sentemce would have sounded perfectly normal, say, 150 years ago. Today it sounds more normal with "de" or "viszont".
Many English speakers might understand this sentence, but it is very stilted English. "Hanem" can't just be translated mechanically as "but" to make a sentence an English speaker will recognize as English. In virtually all of these "hanem" sentences, English makes a separate independent clause: "That popular singer doesn't have a car, she has an airplane." I hope that, before this course moves on from its Beta stage, that the creators will address this serious problem, along with the verbs of standing combined with the adverbs of directional motion (-be áll, etc.).