How does the Celestial Brilliance interact with the Deeper Darkness spell?

I am in doubt about what happens when two area effects that influence a place’s light interact with one another.

Celestial Brilliance (Book of Exalted Deeds, p. 94) is a 4th-level spell and reads, in part:

Celestial brilliance brought into an area of magical darkness (or vice versa) is temporarily negated, so that the otherwise prevailing light conditions exist in the overlapping areas of effect.

Celestial brilliance counters or dispels any darkness spell of equal or lower level, such as deeper darkness.

Does this mean that if I cast Deeper Darkness as a 3rd-level spell within Celestial Brilliance‘s area, then both neutralize?

What I do not understand is if it is a matter of spell level or timing (i.e. which spell is cast first).

Also, is there any difference – other than the damage – between the light produced by Celestial Brilliance and Daylight? The first one says “light brighter than bright sunlight”; the other one says “light as bright as full daylight”. Does this difference in phrasing imply anything?