The core distinction
Illuminance measures incoming light — how much light energy is landing on a given surface area, regardless of what that surface does with it afterward. Luminance measures outgoing light — how much light is actually leaving a surface toward a viewer, whether by emitting it directly (a screen) or reflecting it (a wall, a printed page). A brightly lit room has high illuminance everywhere, but a black object in that room still has low luminance, since it absorbs most of the light rather than reflecting it back.
The units, and how they relate within each concept
Illuminance units: lux (the SI unit) and footcandle (an imperial-derived unit still common in some US contexts) — 500 lux converts to about 46.45 footcandles. Luminance units: candela per square meter (cd/m², the direct SI unit), nit (a fully equivalent name for cd/m² — 100 cd/m² is exactly 100 nit, not just approximately), and footlambert (another imperial-derived unit — 100 cd/m² converts to about 29.19 footlamberts).
Why "nit" is just another name, not a different unit
Unlike footcandle-vs-lux or footlambert-vs-cd/m² (which involve an actual conversion factor between imperial and metric-derived units), nit and candela per square meter are exactly, identically the same unit with two different names — "nit" is simply a shorter, more convenient term used especially in display and screen-brightness specifications (a phone or monitor's brightness spec in nits is the identical quantity as its brightness in cd/m²).
A concrete example showing why both matter separately
A well-lit office might have around 500 lux of illuminance on the desks — a measure of how much light is arriving there. A computer monitor sitting on that desk might have a luminance of around 250-350 nits (a typical office monitor's rated brightness) — a completely separate measurement of how much light the screen itself emits toward the viewer. Increasing the room's illuminance (brighter room lighting) doesn't change the monitor's luminance at all, since the monitor generates its own light rather than reflecting the room's.
Where each measurement is actually used
Illuminance (lux) is what lighting designers specify for a space — office lighting standards, reading light recommendations, outdoor lighting codes are all expressed in lux, since what matters there is how much light reaches the work surface or ground. Luminance (nits or cd/m²) is what matters for anything you look directly at — display brightness specs, glare assessment, and how visible an object or sign appears against its background, since that depends on how much light actually reaches the viewer's eye from that specific surface.