The title kind of explains it. What role does deity worship/interaction play for non-cleric/paladin types? Do deities ever grant boons to non-cleric/paladins that are still devout in worship?
Aside from Roleplaying benefits (the wizard who worship Nethys as the god of magic and knowledge, the pirate swashbuckler who worships Besmara to have the goddess of pirates and the sea on his side, etc...), there are a few mechanical ones (aside from the obvious GM rulings about boons for RP) like the feats Deific, Demonic, and Celestial Obedience:
Each deity (Demon lord, Empyreal Lord) requires a different daily obedience, but all obediences take no more than 1 hour per day to perform. Once you've performed the obedience, you gain the benefit of a special ability or resistance as indicated in the "Obedience" entry for the god to whom you performed the obedience.
You can only do the obedience for a god/demon/empyreal you worship, so that gives a mechanic effect.
Outside of those feats, some prestige classes and some archetypes also require you worship certain gods. The Dissident of Dawn prestige class (even though its a divine caster, I'm including it as an example) and the Bard archetype Dervish of Dawn require you to worship Sarenrae, even though the bard doesn't get divine spellcasting from it.
Though I mentioned it briefly above, Deities coming down and granting boons or sending messengers to do the same are specifically GM related. Your adventure may or may not involve direct interaction with a god; that is left to your GM or the published adventure you're running/playing in.
Another small note is that in theory, some creatures would be far less likely to attack you based on whether you are devout in your worship to a particular deity. There are certain creatures (danged if I can find the references now) that do seek out worshipers of specific gods to cull them, some that will ignore worshipers of specific gods, etc. As well as the obvious fact that you're more likely to get in good with the religion you worship if it has a place in the town(s) you interact with.
Worshipping a deity can reflect on your game in many various ways:
It will give your character some background fluff.
It makes him able to access some feats, traits and prestige classes (see Gnomejon's answer for more details).
Some spells or other capacities have effects in relation to the religion, for example you can set a glyph of warding so it doesn't activate toward someone of the right religion.
The NPCs can reject people of some religion, for example in Rahadoum you better not be too obviously pious.
A god can sometimes help one of his loyal followers. There is no set rules for that, but some GMs can houserule it. For example in a game I played Sarenrae told the priest of the local temple to give a relic to our mage, who was very devout. This can be used as a plot hook and should not give some character power of the scale of what the cleric have, as it would unbalance the game.