Reputation isn’t payment and shouldn’t (usually) be used in place of it
Reputation is a concept designed to help the GM arbitrate the world. Players may appreciate knowing when their Rep moves, but the characters are not being given tokens of +1 with the Theives’ Guild. Most characters expect or at least appreciate payment for service rendered, and continued accrual of valuables is a hallmark of the genre as well as a key component if you follow Paizo’s balance efforts.
This is not to say gratitude can’t be a reward. If the PC’s save an NPC who has nothing to their name, or perform favors too small to rate actual payment, you don’t need to introduce a +1 striking weapon that you hadn’t intended. Also occasionally, groups may wish to perform duties to organizations and decline payment in favor of increasing their standing. That’s OK! I would personally allot an extra Rep point for factions that would appreciate this showing of altruism (and perhaps reduce it for groups that would see it as weakness… although I’m be less tempted by that idea).
This is all most exemplified in the final section of the Reptuation page on AoN, Running Reputation. It is additionally made clear that it also isn’t some ephemeral currency, as it can be thrown away or gained at the whim of the GM.
Using Reputation Points in the background allows you to track the PCs’ status with a variety of groups or factions in your campaign based on the PCs’ actions and choices. This is especially useful for a sandbox campaign or other structure with numerous possibilities, helping you build a reactive structure and track how PCs are doing with groups they interact with only occasionally. When the point totals don’t fit the story, use common sense. For instance, if the PCs are hunted by the town guard due to a corrupt mayor, but then perform a major quest revealing the mayor’s wrongdoing and saving the town, that might jump them from the worst negative reputation to a much more positive reputation. Similarly, the PCs’ foes could frame them, harming the PCs’ reputation though they’ve done no disservice.
