Specifying, Goddess rep, I know what faction rep is for.
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.