The amount of light projected at the screen by a projector is measured in ansi-lumen but we'll simply call it the lumen output. The actual number listed in a projector's spec sheet is about as meaningless as can be as it does not say anything about the quality of the image projected; a 2000-ansi-lumen projector is not better than a 1000-ansi-lumen projector by default. Something that is often forgotten is the simple fact that the quality and contrast of the projected image depends on the amount of ambient light and the size of the image we are projecting.
If we are, for example, projecting in a darkened room where home theatre projectors are intended to be used the 2000-ansi-lumen projector might actually have too much lumen output and create an overly bright and washed out image. Furthermore we must always factor in the size of the projected image. If we, for example, take another look at the 2000-ansi-lumen projector and have it project an image of 2 x 1.5 meters and we enlarge the image size to 4 x 3 meters, four times the surface area, the brightness of the projected image will just be a quarter of the original.
The left projector has too much lumen output, washing out the image, the right one is about right.
So, in a nutshell, the lumen output of a projector is important, but only if you factor in the lighting conditions of the room it will be used in and the size of the image that you will be projecting. In the case of a home theatre, lumen output is not really an important factor, as we are always projecting in a darkened room and at a known image size. Most home theatre setups use a projector with a lumen output of about 300 to 1000 ansi-lumen to not saturate and reduce the contrast of the projected image.
Contrast comparison of two projectors, notice the details in dark areas of the right image.
Contrast is a much more important factor to consider as the contrast ratio of a projector tells us how large the difference is between the projector projecting a black and a white image. Contrast is directly related to the amount of detail we see in dark scenes, a projector with a higher contrast ratio will always project a richer image with better color saturation than a projector with a lower contrast ratio simply because the difference between black and white is greater.
For home theatre the contrast ratio is one of the most important factors because we project in a darkened room. If the projector does not have a sufficient contrast ratio the projector will seem to project black as a dark grey. When watching movies with lots of dark scenes, details will fade away into the gray as the projector is unable to display sufficient contrast levels to bring out the fine details. So the contrast ratio of a projector is very important; a higher contrast ratio will always pay off, under any condition.