If the areas of light and darkness simply intersect, then they temporarily cancel each other in the region of overlap. Once the sources are moved away from each other, both spells keep functioning normally.
If one is intentionally cast to dispel the other, then both effects end, without a caster level check. Say you have haste running, and I hit you with a slow spell. There is no saving throw or level check; the haste spell automatically ends, and that's the only effect of my casting.
Counterspelling also requires no caster level check. If I ready an action to counterspell you, and you cast haste, I can counter it with either another haste or a slow. Both choices work the same way, and your spell fizzles without effect.