yea, I can't stand ftl
The following is my current understanding of luminance and illuminance measurements - I might be off in some details:
The geometry of measuring luminance means that it's a distance independent quantity. In other words, it is a quantity that does not change over distance. So a luminance meter is essentially measuring the luminance of the display, even if you take that measurement from a distance (so long as the entire display is subtended by the measuring angle).
When you measure illuminance, you're measuring the luminous flux per unit area. The area in question is the sensor in the instrument. Thus, the farther away you hold the instrument, the smaller the illuminance.
A luminance meter uses optics so that only a narrow solid angle of light is being measured. See the first diagram on this
page. The important thing to note here (and this is the distance independence idea) is that as the instrument moves further away from the light source, a larger area of the light source will be channeled into the measurement angle. This will perfectly counterbalance the attenuation of light that is due to the increased distance. Note that the sensor is actually measuring illuminance, but it can calculate luminance by dividing the measured illuminance by the measuring angle. And, assuming the luminance of the light source doesn't change appreciably over that small measuring angle, it will give a good approximation to the actual luminance.
An illuminance meter essentially does the same thing, but without optics, and measures over 180 degrees. Because of this, unless you have the meter literally right up against the screen, a substantial portion of the measured light will be coming from beyond the display being measured. This is fine, if all you are interested in is the illuminance at any given point in the room (due to all light sources). More importantly, even if you do have the illuminance meter right up against the screen, the illuminance being measured is due to light coming from more than just a small angle. And unless your display has PERFECT viewing angles (i.e. it is a perfect lambertian radiator, with equal luminance in every direction), then dividing the measured illumiance by 180 is only going to give you an "average" luminance over all these angles.
(note: technically, we should be speaking in solid angle and in terms of the surface of the unit sphere: i.e. steradians). But it's simpler to sometimes explain in terms of degrees in two dimensions.