How can a Hydrogen make such a difference?
The difference in structure of these two compounds is  perhaps the presence of a more conjugated structure in biliverdin compared to bilirubin. These products (degradation products of haeme1), just as most pigments, make use of the conjugated electron systems to absorb visible light, giving rise to strong colors. The long chain conjugated system in bilverdin leads to its strong green color, the reduced biliribin (methenyl bridge) has a less conjugated system with a different absorption consenquently different color.

This is a case of effects of conjugation in pigments than merely a difference in functional groups as I see it. 

1Note The degradation of haeme occurs at  the end of the red cells’  lifetime, where they are removed from the circulation and their components degraded, thus depends of many factors most importantly regulations carried out in the body, enzymes etc.

References


Voet and Voet Biochemistry Section 26-4. Amino Acids as Biosynthetic Precursors.
Conjugated system