From my experience, goddess Rep affects:

1. Faction rep of Factions aligned to the goddess. (For example, increasing Metal rep increases Faction rep with old Guardians)

2. The chance of finding a useable altar to a goddess. (? I believe at least)

3. Enemy factions that can appear when you step on a trap. (With low Metal goddess rep, again as an example, hostile Wolfgirl trainers can appear.)

4. if you break an orb and it would decrease your faction rep below 0% with a goddess, Orb guardians are spawned on the map.

5. Goddess rep gives you a bonus or penalty to struggling against restraints associated with that goddess.