It really comes down to scenes that are fairly dark. Let's consider two projectors - one with good blacks, one with great ones.
On an average scene, perhaps a daytime, no sunlight shot, if there is supposed to be a true black area in the image, perhaps parts of a black car, in the shade, the "good" one will do a very good job. The great one's black car will be a touch blacker, but you probably wouldn't notice unless seen side by side. In other words, the difference would be very slight.
Now switch to a very dark scene, such as the night train scene I use below for shadow detail comparison. All of a sudden the difference between the two projectors is quite dramatic. The lower black levels of the great one would make its image "pop", and the good one's image will seem rather dull by comparison. That's the story - short version.
The darker the scene, the more the blacks matter. Thing is, most movies have some fairly dark scenes, and even a significant percentage have some really dark scenes. Now don't forget that if you've got a 15 watt light on in the room, it would throw enough ambient light, that the difference between projectors would be relatively insignificant, yet still there.