I think I understand your confusion, since that text is quite oddly phrased. However, the key text here is that
Celestial brilliance counters or dispels any darkness spell of equal or lower level, such as deeper darkness.
If Deeper Darkness is cast as a 4th level or lower spell it is countered by Celestial Brilliance, removing the darkness.
However, that top portion of the spell says that Celestial Brilliance is temporarily negated to reveal the otherwise prevailing light conditions. How I interpret this is that if the darkness spell encountering the Celestial Brilliance is unable to be countered, the aura of light from Celestial Brilliance disappears so the area looks like however it would normally look, absent Deeper Darkness or Celestial Brilliance. So if the area was in bright daylight it would be bright, and if it had been in darkness it would still be in darkness.
This means that Celestial Brilliance will always somewhat succeed in countering magical darkness, but in the case where the darkness spell is of a higher level than Celestial Brilliance (5th or higher in this example) the damaging effects of Celestial Brilliance are negated.
The alternative interpretation is that if Celestial Brilliance fails to dispel darkness, it itself is suppressed while within the darkness and ceases to function. The reason I believe that this interpretation is not particularly likely is that the word choice of the spell seems to omit the details necessary to properly explain such an effect. Were the spell to be suppressed fully by a more powerful magical darkness effect I would expect it to say
Celestial Brilliance brought into a magical darkness spell of higher level than Celestial Brilliance is temporarily suppressed, negating all spell effects within that area. Outside of the magical darkness the spell effects continue as before.
Since it doesn't I choose to believe that the two effects negate each other allowing whatever prior conditions existed to display themselves.
As to whether the description of "light brighter than bright sunlight" and "light as bright as full daylight" make a difference, I think in most cases they would not be meaningfully different. However, in any contest, Celestial Brilliance would be brighter than Daylight, due to it being phrased as "brighter than bright sunlight", as compared to "as bright as full daylight" since sunlight and daylight can be understood to be synonyms for the same phenomenon.